Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product

1.   Introduction

Throughout this paper, let ${\mathbb C}^{m \times n}$ denote the collection of all $m \times n$ matrices over the field of complex numbers; $A^{\ast}$ denote the conjugate transpose; $r(A)$ denote the rank of $A$, i.e., the maximum order of the invertible submatrix of $A$; ${\mathscr R}(A) = \{Ax \, | \, x \in {\mathbb C}^{n}\}$ denote the range (column space) of a matrix $A \in {\mathbb C}^{m \times n}$. An $A \in {\mathbb C}^{m \times m}$ is said to be idempotent if $A^2 = A$; to be Hermitian if $A = A^{\ast}$; to be an orthogonal projector if $A^2 = A = A^{\ast}$; to be EP (range-Hermitian) if ${\mathscr R}(A) = {\mathscr R}(A^{\ast})$; to be normal if $AA^{\ast} = A^{\ast}A$. The Moore–Penrose generalized inverse of $A \in {\mathbb C}^{m \times n}$, denoted by $A^{†}$, is the unique matrix $X \in {\mathbb C}^{n \times m}$ satisfying the four Penrose equations

A matrix $X$ is called a $\{i, \ldots, j\}$-generalized inverse of $A$, denoted by $A^{(i, \ldots, j)}$, if it satisfies the $i$th, $\ldots$, $j$th equations in (1.1). The collection of all $\{i, \ldots, j\}$-generalized inverses of $A$ is denoted by $\{A^{(i, \ldots, j)}\}$. There can be found many different kinds of definitions of generalized inverses for a matrix. In comparison, generalized inverses that are defined by the four Penrose equations are most popular and well developed. It can be seen that there are all 15 types of $\{i, \ldots, j\}$-generalized inverses of $A$ by the above definition. All these generalized inverses have rich structures and occur in various theoretical and applied problems, yet $A^{†}$, $A^{(1, 3, 4)}$, $A^{(1, 2, 4)}$, $A^{(1, 2, 3)}$, $A^{(1, 4)}$, $A^{(1, 3)}$, $A^{(1, 2)}$, and $A^{(1)}$ are usually called the eight commonly-used types of generalized inverses of $A$ in the literature. In particular, the Moore–Penrose generalized inverse of a matrix $A$ was specially studied and recognized because $AA^{†}$, $A^{†}A$, $I_m - AA^{†}$, and $I_n - A^{†}A$ are four orthogonal projectors onto the ranges and kernels of $A$ and $A^{\ast}$, respectively. Hence, it optimizes a number of interesting properties, and thus has extraordinary values in theoretical and computational mathematics with applications. For brief, we denote $P_A = AA^{†}$ and $E_{A} = I_m - AA^{†}$ in the sequel. It can also be used to represent other generalized inverses by means of certain algebraic operations of $A$ and $A^{†}$. The four matrix equations in (1.1) were proposed by Sir Penrose in his seminal paper ^[17]. It is obvious that generalized inverses of a matrix are fully determined by the four equations in (1.1), which are direct extensions to singular matrix of the four fundamental inverse operation properties $AA^{-1}A = A$, $A^{-1}AA^{-1} = A^{-1}$, $(AA^{-1})^{\ast} = AA^{-1}$, and $(A^{-1}A)^{\ast} = A^{-1}A$ for a nonsingular matrix $A$, and thus the algebraic connotations and characteristics of generalized inverses of a matrix are consistent with these equalities and their variations when computing and utilizing generalized inverses of matrices under various situations. In fact, generalized inverses, as an important branch of current matrix algebra, have been already developed as an independent theoretical system and analysis method in mathematics and applications. It, however, would still be of great practical and theoretical significance to deepen the study of generalized inverses from the perspective of many conceptual and fundamental problems. We refer the reader to the traditional reference books ^[1,2] for more expositions on generalized inverses of matrices with a century's history.

Notice that generalized inverses of a matrix are defined to be certain common solutions of the four Penrose matrix equations. Hence it is natural to conduct various operations of matrices and generalized inverses and to establish various equalities for matrices and their generalized inverses from theoretical and applied points of view. As usual, the theory of generalized inverses processes its own lists of (more or less celebrated) problems and open questions, sometimes they are hard to appreciate or just to understand if one does not work in this field. We begin with a simple case to illustrate. Assume that $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p}$, and $C \in {\mathbb C}^{p \times q}$. Then the pair of matrix products $AB$ and $ABC$ are defined accordingly. If they are all square matrices of the same size and assume that they are invertible matrices, then the following two best-known matrix equalities for the ordinary inverses of the pair of matrix products

hold, which are usually called the reverse order laws (ROLs) for the ordinary inverses of the matrix products $AB$ and $ABC$, respectively. As fundamental properties of inverse operations of matrices, they can be used to simplify various matrix expressions that involve the inverses of products of matrices. If these matrices and their products are singular, then $A^{†}$, $B^{†}$, $C^{†}$, $(AB)^{†}$ and $(ABC)^{†}$ do exist. In these cases, it is also necessary to describe the relations among these matrices. In particular, it is easy to propose the following two ROLs for the Moore–Penrose generalized inverses of $AB$ and $ABC$:

as natural extensions of the ROLs for the ordinary inverses of matrix products in (1.2). It is not until in 1960s that mathematicians paid attention to the study of ROLs for generalized inverses of matrix products due to the non-commutativity of matrix algebra and the lack of methodologies for dealing with complicated matrix expressions and matrix equalities that involve generalized inverses. Since then, there has been a long-term interest in the research of the two ROLs in (1.3), and there are many classic and profound results that have been obtained on the two ROLs and their variations; see e.g., some earlier and recent work ^{[5,6,8,9,10,12,18,20,25,26,27]}. In addition to these standard ROLs for the Moore–Penrose generalized inverses, it may be useful to examine further other kinds of reasonable expressions composed of generalized inverses of matrix products according to the conventional algebraic operations of matrices. Surprisingly, there exist many kinds of matrix equalities that are composed of nested products of matrices and their generalized inverses. Here we mention a group of well-known nested ROLs:

for the Moore–Penrose generalized inverse of $AB$, and two groups of nested ROLs:

and

for the Moore–Penrose generalized inverse of the triple matrix product $ABC$. It should be pointed out that the common feature of the matrix equalities in (1.4)–(1.21) is: they involve the orthogonal projectors $A^{†}A$, $BB^{†}$, $CC^{†}$, $(AB)^{†}AB$, and $BC(BC)^{†}$ in different places in the nested products of matrices on the right-hand sides, respectively, rather than the straightforward reverse order products $B^{†}A^{†}$ and $C^{†}B^{†}A^{†}$. Therefore, they provide a mixture of multiple expressions of the Moore–Penrose generalized inverses of $AB$ and $ABC$ from the standpoint of orthogonal projectors. On the other hand, they all agree with (1.2) when $A$, $B$, and $C$ are invertible matrices. To understand the motivation and reasonability of (1.4)–(1.21), it is instructive to rewrite $AB$ and $ABC$ as the following multiple matrix products:

according to the definition of the Moore–Penrose generalized inverse of matrix. In these situations, applying (1.3) to the nested products in (1.22) and (1.23), respectively, we obtain (1.4)–(1.6) and (1.10)–(1.12). Equations (1.13)–(1.15) and (1.19)–(1.21) result from the products of the right-hand sides of (1.4), (1.5), (1.10), and (1.11) with $AB$ and $ABC$, respectively, and simplification. Moreover, it is worth noting that the triple matrix product $ABC$ can be written as

which is in fact a special situation of (1.23) by replacing $A$, $B$, and $C$ by $AB$, $B^{†}$, and $BC$, respectively. Applying (1.10)–(1.15) to (1.24) leads to (1.16)–(1.21), respectively. So that there is reasonable discourse for suggesting the ROLs in (1.4)–(1.21). This kind of consideration was first given in ^[5], see also ^{[7,11,21,22,23]} for some similar work. It should be pointed out that nested ROLs in (1.4)–(1.21) are different from the two ROLs in (1.3) and also they are not necessarily equivalent to the two ROLs in (1.3). However, there is solid reason for deriving various necessary and sufficient conditions for all these matrix equalities to hold by means of various usual algebraic operations of matrices. In fact, people approached the ROLs in (1.4)–(1.21) respectively for matrices, as well as elements in rings, Hilbert spaces, and $C^{\ast}$-algebras in the literature, such as, parts of (1.4)–(1.9) were formulated and approached in ^{[3,7,11,14,15,21,22]} and parts of (1.10)–(1.21) were considered in ^{[4,12,23,26,28,29]} among others. It has been realized that ROLs may have millions of reasonable forms and thus they have become one of the fruitful research fields in the matrix algebra.

Because of important roles of generalized inverses in dealing with singular matrices and the reasonability of construction of ROLs, people have paid great attention to the research of ROLs under various general assumptions since 1960s. In fact, ROLs have been being conceptually amongst the most pleasant research topics concerning algebraic equalities composed of generalized inverses of matrices for people to work with. Recall that equalities and equations of matrices can be constructed arbitrarily through various conventional algebraic operations of matrices. But it is not easy to adequately address the challenges identified in determining consistency conditions of a given matrix equation and finding general solutions of the equation under general assumptions. It can be seen from (1.3)–(1.21) that reverse order laws problems are mainly concerned with establishing various reasonable equalities for products of singular matrices and their generalized inverses. Because of the non-commutativity of matrix algebra, (1.3)–(1.21) do not necessarily hold. On the other hand, observe that the right-hand sides of (1.4)–(1.9), and (1.10)–(1.21) have different structures of matrix operations. Hence the matrix equalities in (1.4)–(1.9), and (1.10)–(1.21) are not necessarily equivalent to the two ROLs in (1.3). Thus it is natural to seek conditions under which (1.4)–(1.21) hold, respectively. In the past several decades, people noticed many fundamental facts regarding the equivalences of equalities composed of matrices and their generalized inverses, some of them were discovered during the investigation of reverse order laws for generalized inverses of matrix products. During the formulation and characterization of matrix equalities, people have realized that the rank of matrix (one of the basic concepts in linear algebra, which can be calculated by counting the number of leading 1s in any row-echelon matrix to which a matrix can be carried by row operations) and various traditional algebraic and symbolical methods for calculating ranks of matrices (called the matrix rank method), can well be used to deal with various simple and complicated matrix equalities and matrix set inclusions. Now it is possible to derive identifying conditions for various ROLs for generalized inverses of matrix products to hold by means of various ordinary and effective matrix analysis tools, including the matrix rank method, the matrix range method, and the block matrix method.

A main focus of this paper is to present a full approach to the nested ROLs in (1.10)–(1.21) and their variations. The work contains the preparations of lots of useful formulas and facts on generalized inverses of matrices, and derivations of many necessary and sufficient conditions for the nested ROLs to hold through use of the matrix rank method and the block matrix method. The results obtained show essential equivalences among various matrix equalities that are composed of products of matrices and their generalized inverses. The paper is organized as follows. In Section 2, we present some preliminary formulas and results regarding operations of generalized inverses, ranks and ranges of matrices, as well as two groups of known results on ROLs of generalized inverses of products of two matrices and their variations. In Sections 3 and 4, we present miscellaneous equalities that are composed of the triple matrix products and their generalized inverses, and give detailed derivations of the main results. Conclusions and some open problems on ROLs for generalized inverses of multiple matrix products are given in Section 5.

2.   Some preliminaries

We begin with presenting an assortment of necessary preliminary results, which will be used as tools in the derivations of the main results in the sequel. The formulas and facts in the following eight lemmas are well recognized and scattered in the literatures ^{[1,2,13,16,19,24]} or easy to prove by the definitions of ranks, ranges, and generalized inverses of matrices.

Lemma 2.1. Let $A \in {\mathbb C}^{m \times n}.$ Then the following results hold

Lemma 2.2. Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{m \times k}.$ Then the following results

hold$.$ In particular$,$ the following results hold$.$

${\rm (a)}$ $r[\, A, \, B \, ] = r(A)$ $\Leftrightarrow {\mathscr R}(B)$ $\subseteq {\mathscr R}(A) \Leftrightarrow AA^{†}B = B \Leftrightarrow E_{A}B = 0 \Leftrightarrow r(E_{B}A) =$ $r(A) - r(B)$ $\Leftrightarrow r(P_{A} - P_{B}) = r(A) - r(B).$

${\rm (b)}$ ${\mathscr R}(A) = {\mathscr R}(B) \Leftrightarrow r[A, \, B]$ = $r(A) = r(B) \Leftrightarrow 2r[A, \, B]$ = $r(A) + r(B) \Leftrightarrow 2r[P_{A}P_{B}, \, P_{B}P_{A}]$ = $2r(P_{A}P_{B}) + 2r(P_{B}P_{A}) - r(A) - r(B)$ $\Leftrightarrow AA^{†} = BB^{†}.$

Lemma 2.3. Let $M \in {\mathbb C}^{m \times n}$ and assume that $A, \, B \in \{M^{(2)}\}.$ Then the following rank equality

holds$.$ In consequence$,$

Lemma 2.4. Let $A_1 \in {\mathbb C}^{m \times n_1},$ $A_2 \in {\mathbb C}^{m \times n_2},$ $B_1 \in {\mathbb C}^{m \times p_1},$ and $B_2 \in {\mathbb C}^{m \times p_2},$ and assume that ${\mathscr R}(A_1) = {\mathscr R}(B_1)$ and ${\mathscr R}(A_2) = {\mathscr R}(B_2).$ Then

Lemma 2.5. Let $A \in {\mathbb C}^{m \times n}$ and $G \in {\mathbb C}^{n\times m}.$ Then

Lemma 2.6. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{m \times p},$ $P \in {\mathbb C}^{p \times m},$ and $Q \in {\mathbb C}^{q \times n}.$ Then the following results hold

Lemma 2.7. Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{n \times p},$ $M \in {\mathbb C}^{n \times n}.$ Then the following range equalities

hold$,$ and the following rank equalities hold

Lemma 2.8. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{p\times n},$ $C \in {\mathbb C}^{m\times q}$, and $P, \, Q \in {\mathbb C}^{s\times t}.$ Then the following results hold$.$

${\rm (a)}$ $AB^{\ast} = 0 \Leftrightarrow BA^{\ast} = 0 \Leftrightarrow AB^{†} = 0 \Leftrightarrow BA^{†} = 0.$

${\rm (b)}$ $A^{\ast}C = 0 \Leftrightarrow C^{\ast}A = 0 \Leftrightarrow A^{†}C = 0 \Leftrightarrow C^{†}A = 0.$

${\rm (c)}$ $PQ^{\ast} = 0$ and $Q^{\ast}P = 0 \Rightarrow (P \pm Q)^{†} = P^{†} \pm Q^{†}.$

Lemma 2.9. Let $M \in {\mathbb C}^{m \times n}$ and assume that $P \in {\mathbb C}^{m \times m}$ and $Q \in {\mathbb C}^{n \times n}$ are two orthogonal projectors$,$ and denote $\widehat{P} = I_m - P$ and $\widehat{Q} = I_n - Q.$ Then the following matrix equalities hold

and the following matrix equalities hold

Proof. It is easy to verify that

hold by expanding the left-hand sides of (2.51)–(2.54). In these cases, applying Lemma 2.8 (c) to both sides of (2.51)–(2.54) yields (2.43)–(2.46). Pre- and post-multiplying both sides of (2.43)–(2.46) with $Q$ and $P$, respectively, and simplifying lead to (2.47)–(2.50).

Finally, we give a summary of known solutions (with some modifications) established in ^[27] to the two nested ROLs in (1.4) and (1.5), which will directly be used to derive the main results in Section 3.

Lemma 2.10. Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{n \times p}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(AB)^{(1, 2, 3)}\} \ni (A^{†}AB)^{†}A^{†}.$

$\langle 2\rangle$ $\{(AB)^{(1, 2, 3)}\} \ni (A^{\ast}AB)^{†}A^{\ast}.$

$\langle 3\rangle$ $\{(A^{†}AB)^{(1, 2, 3)}\} \ni (AB)^{†}A.$

$\langle 4\rangle$ $\{(A^{\ast}AB)^{(1, 2, 3)}\} \ni (AB)^{†}(A^{†})^{\ast}.$

$\langle 5\rangle$ $\{(AB)^{(1, 2, 3)}\} \ni B^{†}(A^{†}ABB^{†})^{†}A^{†}.$

$\langle 6\rangle$ $\{(AB)^{(1, 2, 3)}\} \ni B^{\ast}(A^{\ast}ABB^{\ast})^{†}A^{\ast}.$

$\langle 7\rangle$ $\{(A^{†}ABB^{†})^{(1, 2, 3)}\} \ni B(AB)^{†}A.$

$\langle 8\rangle$ $\{(A^{\ast}ABB^{\ast})^{(1, 2, 3)}\} \ni (B^{†})^{\ast}(AB)^{†}(A^{†})^{\ast}.$

$\langle 9\rangle$ $(AB)^{†} = (A^{†}AB)^{†}A^{†}.$

$\langle 10\rangle$ $AB = ((A^{†}AB)^{†}A^{†})^{†}.$

$\langle 11\rangle$ $AB = A((AB)^{†}A)^{†}.$

$\langle 12\rangle$ $(AB)^{†}A = (A^{†}AB)^{†}.$

$\langle 13\rangle$ $B(AB)^{†}A = B(A^{†}AB)^{†}.$

$\langle 14\rangle$ $AB(AB)^{†} = AB(A^{†}AB)^{†}A^{†}.$

$\langle 15\rangle$ $AB(AB)^{†}A = AB(A^{†}AB)^{†}.$

$\langle 16\rangle$ $B^{†}A^{†}AB(AB)^{†} = B^{†}A^{†}.$

$\langle 17\rangle$ $(B^{†}A^{†})^{†}B^{†}A^{†}AB = AB.$

$\langle 18\rangle$ $AB(AB)^{†} = (B^{†}A^{†})^{†}B^{†}A^{†}.$

$\langle 19\rangle$ $(AB)^{†} = (A^{\ast}AB)^{†}A^{\ast}.$

$\langle 20\rangle$ $AB = ((A^{\ast}AB)^{†}A^{\ast})^{†}.$

$\langle 21\rangle$ $AB = (A^{†})^{\ast}((AB)^{†}(A^{†})^{\ast})^{†}.$

$\langle 22\rangle$ $(AB)^{†}(A^{†})^{\ast} = (A^{\ast}AB)^{†}.$

$\langle 23\rangle$ $B(AB)^{†}(A^{†})^{\ast} = B(A^{\ast}AB)^{†}.$

$\langle 24\rangle$ $AB(AB)^{†} = AB(A^{\ast}AB)^{†}A^{\ast}.$

$\langle 25\rangle$ $AB(AB)^{†}(A^{†})^{\ast} = AB(A^{\ast}AB)^{†}.$

$\langle 26\rangle$ $(ABB^{†}A^{†})^{†} = (B^{†}A^{†})^{†}(AB)^{†}.$

$\langle 27\rangle$ $AB(A^{†}AB)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 28\rangle$ $AB(A^{\ast}AB)^{†}A^{\ast}$ is an orthogonal projector$.$

$\langle 29\rangle$ $A^{†}(AB)(AB)^{†}A$ is an orthogonal projector$.$

$\langle 30\rangle$ $A(A^{†}ABB^{†})^{†}A^{†}$ is an orthogonal projector$.$

$\langle 31\rangle$ $ABB^{\ast}(A^{\ast}ABB^{\ast})^{†}A^{\ast}$ is an orthogonal projector$.$

$\langle 32\rangle$ $AB(AB)^{†}$ and $AA^{\ast}$ commute$.$

$\langle 33\rangle$ $AB(A^{†}AB)^{†}A^{†}$ and $AA^{\ast}$ commute$.$

$\langle 34\rangle$ $AB(A^{\ast}AB)^{†}A^{\ast}$ and $AA^{\ast}$ commute$.$

$\langle 35\rangle$ $A^{†}(AB)(AB)^{†}A$ and $A^{\ast}A$ commute$.$

$\langle 36\rangle$ $ABB^{†}A^{†}$ is EP$.$

$\langle 37\rangle$ $\{((A^{†})^{\ast}B)^{(1, 2, 3)}\} \ni (A^{†}AB)^{†}A^{\ast}.$

$\langle 38\rangle$ $\{((A^{†})^{\ast}B)^{(1, 2, 3)}\} \ni ((A^{\ast}A)^{†}B)^{†}A^{†}.$

$\langle 39\rangle$ $\{(A^{†}AB)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}.$

$\langle 40\rangle$ $\{((A^{\ast}A)^{†}B)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}B)^{†}A.$

$\langle 41\rangle$ $\{((A^{†})^{\ast}B)^{(1, 2, 3)}\} \ni B^{†}(A^{†}ABB^{†})^{†}A^{\ast}.$

$\langle 42\rangle$ $\{((A^{†})^{\ast}B)^{(1, 2, 3)}\}$ $\ni B^{\ast}((A^{\ast}A)^{†}$ $BB^{\ast})^{†}A^{†}.$

$\langle 43\rangle$ $\{(A^{†}ABB^{†})^{(1, 2, 3)}\}$ $\ni B((A^{†})^{\ast}B)^{†}$ $(A^{†})^{\ast}.$

$\langle 44\rangle$ $\{((A^{\ast}A)^{†}BB^{\ast})^{(1, 2, 3)}\}$ $\ni (B^{†})^{\ast}$ $((A^{†})^{\ast}B)^{†}A.$

$\langle 45\rangle$ $((A^{†})^{\ast}B)^{†} =$ $(A^{†}AB)^{†}A^{\ast}.$

$\langle 46\rangle$ $((A^{†})^{\ast}B)^{†} =$$((A^{\ast}A)^{†}B)^{†}A^{†}.$

$\langle 47\rangle$ $(A^{†}AB)^{†} = ((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}.$

$\langle 48\rangle$ $((A^{\ast}A)^{†}B)^{†} =$$((A^{†})^{\ast}B)^{†}A.$

$\langle 49\rangle$ $A(B^{†}A^{†}A)^{†}B^{†}A^{†}$ is an orthogonal projector$.$

$\langle 50\rangle$ $(A^{†})^{\ast}(B^{†}(A^{\ast}A)^{†})^{†}B^{†}A^{†}$ is an orthogonal projector$.$

$\langle 51\rangle$ $A^{†}(B^{†}A^{†})^{†}(B^{†}A^{†})A$ is an orthogonal projector$.$

$\langle 52\rangle$ $A(BB^{†}A^{†}A)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 53\rangle$ $(A^{†})^{\ast}(BB^{\ast})^{†}((A^{\ast}A)^{†}(BB^{\ast})^{†})^{†}A^{†}$ is an orthogonal projector$.$

$\langle 54\rangle$ $(B^{†}A^{†})^{†}(B^{†}A^{†})$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 55\rangle$ $A(B^{†}A^{†}A)^{†}B^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 56\rangle$ $(A^{†})^{\ast}(B^{†}(A^{\ast}A)^{†})^{†}B^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 57\rangle$ $A^{†}(B^{†}A^{†})^{†}(B^{†}A^{†})A$ and $(A^{\ast}A)^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}(AA^{\ast}AB) = {\mathscr R}(AB).$

$\langle 59\rangle$ ${\mathscr R}((A^{†})^{\ast}(B^{†})^{\ast}) = {\mathscr R}(AB).$

$\langle 60\rangle$ $r[AA^{\ast}AB, \, AB] = r(AB).$

$\langle 61\rangle$ $r[(A^{†})^{\ast}(B^{†})^{\ast}, \, AB] = r(AB).$

$\langle 62\rangle$ $r[(A^{\ast}AA^{\ast})^{†}B, \, (A^{†})^{\ast}B] = r((A^{†})^{\ast}B).$

Lemma 2.11. Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{n \times p}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(AB)^{(1, 2, 4)}\} \ni B^{†}(ABB^{†})^{†}.$

$\langle 2\rangle$ $\{(AB)^{(1, 2, 4)}\} \ni B^{\ast}(ABB^{\ast})^{†}.$

$\langle 3\rangle$ $\{(ABB^{†})^{(1, 2, 4)}\} \ni B(AB)^{†}.$

$\langle 4\rangle$ $\{(ABB^{\ast})^{(1, 2, 4)}\} \ni (B^{†})^{\ast}(AB)^{†}.$

$\langle 5\rangle$ $\{(AB)^{(1, 2, 4)}\} \ni B^{†}(A^{†}ABB^{†})^{†}A^{†}.$

$\langle 6\rangle$ $\{(AB)^{(1, 2, 4)}\} \ni B^{\ast}(A^{\ast}ABB^{\ast})^{†}A^{\ast}.$

$\langle 7\rangle$ $\{(A^{†}ABB^{†})^{(1, 2, 4)}\} \ni B(AB)^{†}A.$

$\langle 8\rangle$ $\{(A^{\ast}ABB^{\ast})^{(1, 2, 4)}\} \ni (B^{†})^{\ast}(AB)^{†}(A^{†})^{\ast}.$

$\langle 9\rangle$ $(AB)^{†} = B^{†}(ABB^{†})^{†}.$

$\langle 10\rangle$ $AB = (B^{†}(ABB^{†})^{†})^{†}.$

$\langle 11\rangle$ $AB = (B(AB)^{†})^{†}B.$

$\langle 12\rangle$ $B(AB)^{†} = (ABB^{†})^{†}.$

$\langle 13\rangle$ $B(AB)^{†}A = (ABB^{†})^{†}A.$

$\langle 14\rangle$ $(AB)^{†}AB = B^{†}(ABB^{†})^{†}AB.$

$\langle 15\rangle$ $B(AB)^{†}AB = (ABB^{†})^{†}AB.$

$\langle 16\rangle$ $(AB)^{†}ABB^{†}A^{†} = B^{†}A^{†}.$

$\langle 17\rangle$ $ABB^{†}A^{†}(B^{†}A^{†})^{†} = AB.$

$\langle 18\rangle$ $(AB)^{†}AB = B^{†}A^{†}(B^{†}A^{†})^{†}.$

$\langle 19\rangle$ $(AB)^{†} = B^{\ast}(ABB^{\ast})^{†}.$

$\langle 20\rangle$ $AB = (B^{\ast}(ABB^{\ast})^{†})^{†}.$

$\langle 21\rangle$ $AB = ((B^{†})^{\ast}(AB)^{†})^{†}(B^{†})^{\ast}.$

$\langle 22\rangle$ $(B^{†})^{\ast}(AB)^{†} = (ABB^{\ast})^{†}.$

$\langle 23\rangle$ $(B^{†})^{\ast}(AB)^{†}A = (ABB^{\ast})^{†}A.$

$\langle 24\rangle$ $(AB)^{†}AB = B^{\ast}(ABB^{\ast})^{†}AB.$

$\langle 25\rangle$ $(B^{†})^{\ast}(AB)^{†}AB = (ABB^{\ast})^{†}AB.$

$\langle 26\rangle$ $(B^{†}A^{†}AB)^{†} = (AB)^{†}(B^{†}A^{†})^{†}.$

$\langle 27\rangle$ $B^{†}(ABB^{†})^{†}AB$ is an orthogonal projector$.$

$\langle 28\rangle$ $B^{\ast}(ABB^{\ast})^{†}AB$ is an orthogonal projector$.$

$\langle 29\rangle$ $B(AB)^{†}(AB)B^{†}$ is an orthogonal projector$.$

$\langle 30\rangle$ $B^{†}(A^{†}ABB^{†})^{†}B$ is an orthogonal projector$.$

$\langle 31\rangle$ $B^{\ast}(A^{\ast}ABB^{\ast})^{†}A^{\ast}AB$ is an orthogonal projector$.$

$\langle 32\rangle$ $(AB)^{†}AB$ and $B^{\ast}B$ commute$.$

$\langle 33\rangle$ $B^{†}(ABB^{†})^{†}AB$ and $B^{\ast}B$ commute$.$

$\langle 34\rangle$ $B^{\ast}(ABB^{\ast})^{†}AB$ and $B^{\ast}B$ commute$.$

$\langle 35\rangle$ $B(AB)^{†}(AB)B^{†}$ and $BB^{\ast}$ commute$.$

$\langle 36\rangle$ $B^{†}A^{†}AB$ is EP$.$

$\langle 37\rangle$ $\{(A(B^{†})^{\ast})^{(1, 2, 4)}\} \ni B^{\ast}(ABB^{†})^{†}.$

$\langle 38\rangle$ $(A(B^{†})^{\ast})^{(1, 2, 4)} \ni B^{†}(A(BB^{\ast})^{†})^{†}.$

$\langle 39\rangle$ $\{(ABB^{†})^{(1, 2, 4)}\} \ni (B^{†})^{\ast}(A(B^{†})^{\ast})^{†}.$

$\langle 40\rangle$ $\{(A(BB^{\ast})^{†})^{(1, 2, 4)}\} \ni B(A(B^{†})^{\ast})^{†}.$

$\langle 41\rangle$ $\{(A(B^{†})^{\ast})^{(1, 2, 4)}\} \ni B^{\ast}(A^{†}ABB^{†})^{†}A^{†}.$

$\langle 42\rangle$ $\{(A(B^{†})^{\ast})^{(1, 2, 4)}\} \ni B^{†}(A^{\ast}A(BB^{\ast})^{†})^{†}A^{\ast}.$

$\langle 43\rangle$ $\{(A^{†}ABB^{†})^{(1, 2, 4)}\} \ni (B^{†})^{\ast}(A(B^{†})^{\ast})^{†}A.$

$\langle 44\rangle$ $\{(A^{\ast}A(BB^{\ast})^{†})^{(1, 2, 4)}\} \ni B(A(B^{†})^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 45\rangle$ $(A(B^{†})^{\ast})^{†} = B^{\ast}(ABB^{†})^{†}.$

$\langle 46\rangle$ $(A(B^{†})^{\ast}) = B^{†}(A(BB^{\ast})^{†})^{†}.$

$\langle 47\rangle$ $(ABB^{†})^{†} = (B^{†})^{\ast}(A(B^{†})^{\ast})^{†}.$

$\langle 48\rangle$ $(A(B^{†})^{\ast})^{†} = B(A(BB^{\ast})^{†})^{†}.$

$\langle 49\rangle$ $B^{†}A^{†}(BB^{†}A^{†})^{†}B$ is an orthogonal projector$.$

$\langle 50\rangle$ $B^{†}A^{†}((BB^{\ast})^{†}A^{†})^{†}(B^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 51\rangle$ $B(B^{†}A^{†})(B^{†}A^{†})^{†}B^{†}$ is orthogonal projector$.$

$\langle 52\rangle$ $B^{†}(BB^{†}A^{†}A)^{†}B$ is an orthogonal projector$.$

$\langle 53\rangle$ $B^{†}((A^{\ast}A)^{†}(BB^{\ast})^{†})^{†}(A^{\ast}A)^{†}(B^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 54\rangle$ $(B^{†}A^{†})(B^{†}A^{†})^{†}$ and $(B^{\ast}B)^{†}$ commute$.$

$\langle 55\rangle$ $B^{†}A^{†}(BB^{†}A^{†})^{†}B$ and $(B^{\ast}B)^{†}$ commute$.$

$\langle 56\rangle$ $B^{†}A^{†}((BB^{\ast})^{†}A^{†})^{†}(B^{†})^{\ast}$ and $(B^{\ast}B)^{†}$ commute$.$

$\langle 57\rangle$ $B(B^{†}A^{†})(B^{†}A^{†})^{†}B^{†}$ and $(BB^{\ast})^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}(B^{\ast}BB^{\ast}A^{\ast}) = {\mathscr R}(B^{\ast}A^{\ast}).$

$\langle 59\rangle$ ${\mathscr R}(B^{†}A^{†}) = {\mathscr R}(B^{\ast}A^{\ast}).$

$\langle 60\rangle$ $r[B^{\ast}BB^{\ast}A^{\ast}, \, B^{\ast}A^{\ast}] = r(AB).$

$\langle 61\rangle$ $r[B^{†}A^{†}, \, B^{\ast}A^{\ast}] = r(AB).$

$\langle 62\rangle$ $r[(BB^{\ast}B)^{†}A^{†}, \, B^{†}A^{†}] = r(B^{†}A^{†}).$

3.   Main results I

In this section, we are primarily concerned with the nested ROLs in (1.10)–(1.15). To begin, we present a group of known results in ^[26] on ROLs for $\{1\}$- and $\{1, 2\}$-generalized inverses of matrices associated with (1.10)–(1.15).

Lemma 3.1. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$

$\langle 1\rangle$ The following 3 matrix set inclusions always hold

$\langle 2\rangle$ The following 3 matrix set inclusions always hold

$\langle 3\rangle$ The following 4 results always hold

$\langle 4\rangle$ The two matrix equalities in (1.13) and (1.14) always hold$.$ In particular$,$ if $A^{†}ABC = BC$ and $ABCC^{†} = AB,$ then $(ABC)^{†} = (BC)^{†}B(AB)^{†}.$

The correctness of (3.1)–(3.10) can directly be verified by the definitions of the generalized inverses and ordinary operations of the given matrices. Given Lemmas 2.10 and 2.11, we are now ready to establish a wide coverage of necessary and sufficient conditions for (1.10) and (1.11) to hold, respectively.

Theorem 3.2. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (A^{†}ABC)^{†}A^{†}.$

$\langle 2\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (A^{\ast}ABC)^{†}A^{\ast}.$

$\langle 3\rangle$ $\{(A^{†}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}A.$

$\langle 4\rangle$ $\{(A^{\ast}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}(A^{†})^{\ast}.$

$\langle 5\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{†}(A^{†}ABC(BC)^{†})^{†}A^{†}.$

$\langle 6\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{\ast}(A^{\ast}ABC(BC)^{\ast})^{†}A^{\ast}.$

$\langle 7\rangle$ $\{(A^{†}ABC(BC)^{†})^{(1, 2, 3)}\} \ni BC(ABC)^{†}A.$

$\langle 8\rangle$ $\{(A^{\ast}ABC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast}.$

$\langle 9\rangle$ $(ABC)^{†} = (A^{†}ABC)^{†}A^{†}.$

$\langle 10\rangle$ $ABC = ((A^{†}ABC)^{†}A^{†})^{†}.$

$\langle 11\rangle$ $ABC = A((ABC)^{†}A)^{†}.$

$\langle 12\rangle$ $(ABC)^{†}A = (A^{†}ABC)^{†}.$

$\langle 13\rangle$ $BC(ABC)^{†}A = BC(A^{†}ABC)^{†}.$

$\langle 14\rangle$ $ABC(ABC)^{†} = ABC(A^{†}ABC)^{†}A^{†}.$

$\langle 15\rangle$ $ABC(ABC)^{†}A = ABC(A^{†}ABC)^{†}.$

$\langle 16\rangle$ $(BC)^{†}A^{†}ABC(ABC)^{†} = (BC)^{†}A^{†}.$

$\langle 17\rangle$ $((BC)^{†}A^{†})^{†}(BC)^{†}A^{†}ABC = ABC.$

$\langle 18\rangle$ $ABC(ABC)^{†} = ((BC)^{†}A^{†})^{†}(BC)^{†}A^{†}.$

$\langle 19\rangle$ $(ABC)^{†} = (A^{\ast}ABC)^{†}A^{\ast}.$

$\langle 20\rangle$ $ABC = ((A^{\ast}ABC)^{†}A^{\ast})^{†}.$

$\langle 21\rangle$ $ABC = (A^{†})^{\ast}((ABC)^{†}(A^{†})^{\ast})^{†}.$

$\langle 22\rangle$ $(ABC)^{†}(A^{†})^{\ast} = (A^{\ast}ABC)^{†}.$

$\langle 23\rangle$ $BC(ABC)^{†}(A^{†})^{\ast} = BC(A^{\ast}ABC)^{†}.$

$\langle 24\rangle$ $ABC(ABC)^{†} = ABC(A^{\ast}ABC)^{†}A^{\ast}.$

$\langle 25\rangle$ $ABC(ABC)^{†}(A^{†})^{\ast} = ABC(A^{\ast}ABC)^{†}.$

$\langle 26\rangle$ $(ABC(BC)^{†}A^{†})^{†} = ((BC)^{†}A^{†})^{†}(ABC)^{†}.$

$\langle 27\rangle$ $ABC(A^{†}ABC)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 28\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ is an orthogonal projector$.$

$\langle 29\rangle$ $A^{†}(ABC)(ABC)^{†}A$ is an orthogonal projector$.$

$\langle 30\rangle$ $A(A^{†}ABC(BC)^{†})^{†}A^{†}$ is an orthogonal projector$.$

$\langle 31\rangle$ $ABC(BC)^{\ast}(A^{\ast}ABC(BC)^{\ast})^{†}A^{\ast}$ is an orthogonal projector$.$

$\langle 32\rangle$ $ABC(ABC)^{†}$ and $AA^{\ast}$ commute$.$

$\langle 33\rangle$ $ABC(A^{†}ABC)^{†}A^{†}$ and $AA^{\ast}$ commute$.$

$\langle 34\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ and $AA^{\ast}$ commute$.$

$\langle 35\rangle$ $A^{†}(ABC)(ABC)^{†}A$ and $A^{\ast}A$ commute$.$

$\langle 36\rangle$ $ABC(BC)^{†}A^{†}$ is EP$.$

$\langle 37\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (A^{†}ABC)^{†}A^{\ast}.$

$\langle 38\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni ((A^{\ast}A)^{†}BC)^{†}A^{†}.$

$\langle 39\rangle$ $\{(A^{†}ABC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}.$

$\langle 40\rangle$ $\{((A^{\ast}A)^{†}BC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}A.$

$\langle 41\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{†}((AB)^{†}ABC(B^{†}BC)^{†})^{†}(AB)^{\ast}.$

$\langle 42\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{\ast}((A^{\ast}A)^{†}BC(BC)^{\ast})^{†}A^{†}.$

$\langle 43\rangle$ $\{(A^{†}ABC(BC)^{†})^{(1, 2, 3)}\} \ni BC((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}.$

$\langle 44\rangle$ $\{((A^{\ast}A)^{†}BC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}((A^{†})^{\ast}BC)^{†}A.$

$\langle 45\rangle$ $((A^{†})^{\ast}BC)^{†} = (A^{†}ABC)^{†}A^{\ast}.$

$\langle 46\rangle$ $((A^{†})^{\ast}BC)^{†} = ((A^{\ast}A)^{†}BC)^{†}A^{†}.$

$\langle 47\rangle$ $(A^{†}ABC)^{†} = ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}.$

$\langle 48\rangle$ $((A^{\ast}A)^{†}BC)^{†} = ((A^{†})^{\ast}BC)^{†}A.$

$\langle 49\rangle$ $A((BC)^{†}A^{†}A)^{†}(BC)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 50\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 51\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ is an orthogonal projector$.$

$\langle 52\rangle$ $A(BC(BC)^{†}A^{†}A)^{†}A^{†}$ is an orthogonal projector$.$

$\langle 53\rangle$ $(A^{†})^{\ast}(BC(BC)^{\ast})^{†}((A^{\ast}A)^{†}(BC(BC)^{\ast})^{†})^{†}A^{†}$ is an orthogonal projector$.$

$\langle 54\rangle$ $((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 55\rangle$ $A((BC)^{†}A^{†}A)^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 56\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$.$

$\langle 57\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ and $(A^{\ast}A)^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}(AA^{\ast}ABC) = {\mathscr R}(ABC).$

$\langle 59\rangle$ ${\mathscr R}((A^{†})^{\ast}BC) = {\mathscr R}(ABC).$

$\langle 60\rangle$ $r[AA^{\ast}ABC, \, ABC] = r(ABC).$

$\langle 61\rangle$ $r[(A^{†})^{\ast}BC, \, ABC] = r(ABC).$

$\langle 62\rangle$ $r[(A^{\ast}AA^{\ast})^{†}BC, \, (A^{†})^{\ast}BC] = r((A^{†})^{\ast}BC).$

Proof. It follows immediately from replacing $B$ by $BC$ in Lemma 2.10.

Theorem 3.3. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}(ABCC^{†})^{†}.$

$\langle 2\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 3\rangle$ $\{(ABCC^{†})^{(1, 2, 4)}\} \ni C(ABC)^{†}.$

$\langle 4\rangle$ $\{(ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}.$

$\langle 5\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}((AB)^{†}ABCC^{†})^{†}(AB)^{†}.$

$\langle 6\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{\ast}ABCC^{\ast})^{†}(AB)^{\ast}.$

$\langle 7\rangle$ $\{((AB)^{†}ABCC^{†})^{(1, 2, 4)}\} \ni C(ABC)^{†}AB.$

$\langle 8\rangle$ $\{(AB)^{\ast}ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast}.$

$\langle 9\rangle$ $(ABC)^{†} = C^{†}(ABCC^{†})^{†}.$

$\langle 10\rangle$ $ABC = (C^{†}(ABCC^{†})^{†})^{†}.$

$\langle 11\rangle$ $ABC = (C(ABC)^{†})^{†}C.$

$\langle 12\rangle$ $C(ABC)^{†} = (ABCC^{†})^{†}.$

$\langle 13\rangle$ $C(ABC)^{†}AB = (ABCC^{†})^{†}AB.$

$\langle 14\rangle$ $(ABC)^{†}ABC = C^{†}(ABCC^{†})^{†}ABC.$

$\langle 15\rangle$ $C(ABC)^{†}ABC = (ABCC^{†})^{†}ABC.$

$\langle 16\rangle$ $(ABC)^{†}ABCC^{†}(AB)^{†} = C^{†}(AB)^{†}.$

$\langle 17\rangle$ $ABCC^{†}(AB)^{†}(C^{†}(AB)^{†})^{†} = ABC.$

$\langle 18\rangle$ $(ABC)^{†}ABC = C^{†}(AB)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 19\rangle$ $(ABC)^{†} = C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 20\rangle$ $ABC = (C^{\ast}(ABCC^{\ast})^{†})^{†}.$

$\langle 21\rangle$ $ABC = ((C^{†})^{\ast}(ABC)^{†})^{†}(C^{†})^{\ast}.$

$\langle 22\rangle$ $(C^{†})^{\ast}(ABC)^{†} = (ABCC^{\ast})^{†}.$

$\langle 23\rangle$ $(C^{†})^{\ast}(ABC)^{†}AB = (ABCC^{\ast})^{†}AB.$

$\langle 24\rangle$ $(ABC)^{†}ABC = C^{\ast}(ABCC^{\ast})^{†}ABC.$

$\langle 25\rangle$ $(C^{†})^{\ast}(ABC)^{†}ABC = (ABCC^{\ast})^{†}ABC.$

$\langle 26\rangle$ $(C^{†}(AB)^{†}ABC)^{†} = (ABC)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 27\rangle$ $C^{†}(ABCC^{†})^{†}ABC$ is an orthogonal projector$.$

$\langle 28\rangle$ $C^{\ast}(ABCC^{\ast})^{†}ABC$ is an orthogonal projector$.$

$\langle 29\rangle$ $C(ABC)^{†}(ABC)C^{†}$ is an orthogonal projector$.$

$\langle 30\rangle$ $C^{†}((AB)^{†}ABCC^{†})^{†}C$ is an orthogonal projector$.$

$\langle 31\rangle$ $C^{\ast}((AB)^{\ast}ABCC^{\ast})^{†}(AB)^{\ast}ABC$ is an orthogonal projector$.$

$\langle 32\rangle$ $(ABC)^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 33\rangle$ $C^{†}(ABCC^{†})^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 34\rangle$ $C^{\ast}(ABCC^{\ast})^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 35\rangle$ $C(ABC)^{†}(ABC)C^{†}$ and $CC^{\ast}$ commute$.$

$\langle 36\rangle$ $C^{†}(AB)^{†}ABC$ is EP$.$

$\langle 37\rangle$ $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}(ABCC^{†})^{†}.$

$\langle 38\rangle$ $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 39\rangle$ $\{(ABCC^{†})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 40\rangle$ $\{(AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}.$

$\langle 41\rangle$ $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{†}ABCC^{†})^{†}(AB)^{†}.$

$\langle 42\rangle$ $\{((AB)^{\ast}AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(AB(CC^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 43\rangle$ $\{((AB)^{†}ABCC^{†})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}AB.$

$\langle 44\rangle$ $\{(AB)^{\ast}AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}((AB)^{†})^{\ast}.$

$\langle 45\rangle$ $(AB(C^{†})^{\ast})^{†} = C^{\ast}(ABCC^{†})^{†}.$

$\langle 46\rangle$ $(AB(C^{†})^{\ast}) = C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 47\rangle$ $(ABCC^{†})^{†} = (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 48\rangle$ $(AB(C^{†})^{\ast})^{†} = C(AB(CC^{\ast})^{†})^{†}.$

$\langle 49\rangle$ $C^{†}(AB)^{†}(CC^{†}(AB)^{†})^{†}C$ is an orthogonal projector$.$

$\langle 50\rangle$ $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 51\rangle$ $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ is an orthogonal projector$.$

$\langle 52\rangle$ $C^{†}(CC^{†}(AB)^{†}AB)^{†}C$ is an orthogonal projector$.$

$\langle 53\rangle$ $C^{†}(((AB)^{\ast}AB)^{†}(CC^{\ast})^{†})^{†}((AB)^{\ast}AB)^{†}(C^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 54\rangle$ $(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 55\rangle$ $C^{†}(AB)^{†}(CC^{†}(AB)^{†})^{†}C$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 56\rangle$ $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 57\rangle$ $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ and $(CC^{\ast})^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}(C^{\ast}CC^{\ast}(AB)^{\ast}) = {\mathscr R}((ABC)^{\ast}).$

$\langle 59\rangle$ ${\mathscr R}(C^{†}(AB)^{†}) = {\mathscr R}(C^{\ast}(AB)^{\ast}).$

$\langle 60\rangle$ $r[(ABCC^{\ast}C)^{\ast}, \, (ABC)^{\ast}] = r(ABC).$

$\langle 61\rangle$ $r[C^{†}(AB)^{†}, \, C^{\ast}(AB)^{\ast}] = r(ABC).$

$\langle 62\rangle$ $r[(CC^{\ast}C)^{†}(AB)^{†}, \, C^{†}(AB)^{†}] = r(C^{†}(AB)^{†}).$

Proof. It follows immediately from replacing $A$ by $AB$ and $B$ by $C$ in Lemma 2.11.

We next derive a family of statements that are equivalent to the nested ROL in (1.12).

Theorem 3.4. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 137 statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = C^{†}(A^{†}ABCC^{†})^{†}A^{†}.$

$\langle 2\rangle$ $C(ABC)^{†}A = (A^{†}ABCC^{†})^{†}.$

$\langle 3\rangle$ $A(C(ABC)^{†}A)^{†}C = ABC.$

$\langle 4\rangle$ $C(A(C(ABC)^{†}A)^{†}C)^{†}A = C(ABC)^{†}A.$

$\langle 5\rangle$ $A^{†}ABCC^{†} = A^{†}(C^{†}(A^{†}ABCC^{†})^{†}A^{†})^{†}C^{†}.$

$\langle 6 \rangle$ $(ABC)^{†} = C^{†}(ABCC^{†})^{†}ABC(A^{†}ABC)^{†}A^{†}.$

$\langle 7\rangle$ $C(ABC)^{†}A = (ABCC^{†})^{†}ABC(A^{†}ABC)^{†}.$

$\langle 8\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (A^{†}ABC)^{†}A^{†}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}(ABCC^{†})^{†}.$

$\langle 9\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (A^{\ast}ABC)^{†}A^{\ast}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 10\rangle$ $\{(A^{†}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}A$ and $\{(ABCC^{†})^{(1, 2, 4)}\} \ni C(ABC)^{†}.$

$\langle 11\rangle$ $\{(A^{\ast}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}(A^{†})^{\ast}$ and $\{(ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}.$

$\langle 12\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{†}(A^{†}ABC(BC)^{†})^{†}A^{†}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}((AB)^{†}ABCC^{†})^{†}(AB)^{†}.$

$\langle 13\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{\ast}(A^{\ast}ABC(BC)^{\ast})^{†}A^{\ast}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{\ast}ABCC^{\ast})^{†}(AB)^{\ast}.$

$\langle 14\rangle$ $\{(A^{†}ABC(BC)^{†})^{(1, 2, 3)}\} \ni BC(ABC)^{†}A$ and $\{((AB)^{†}ABCC^{†})^{(1, 2, 4)}\} \ni C(ABC)^{†}AB.$

$\langle 15\rangle$ $\{(A^{\ast}ABC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast}$ and $\{(A^{\ast}ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast}.$

$\langle 16\rangle$ $(ABC)^{†} = (A^{†}ABC)^{†}A^{†}$ and $(ABC)^{†} = C^{†}(ABCC^{†})^{†}.$

$\langle 17\rangle$ $(A^{†}ABC)^{†} = (ABC)^{†}A$ and $(ABCC^{†})^{†} = C(ABC)^{†}.$

$\langle 18\rangle$ $ABC(ABC)^{†} = ABC(A^{†}ABC)^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}(ABCC^{†})^{†}ABC.$

$\langle 19\rangle$ $ABC(ABC)^{†}A = ABC(A^{†}ABC)^{†}$ and $C(ABC)^{†}ABC = (ABCC^{†})^{†}ABC.$

$\langle 20\rangle$ $(BC)^{†}A^{†}ABC(ABC)^{†} = (BC)^{†}A^{†}$ and $(ABC)^{†}ABCC^{†}(AB)^{†} = C^{†}(AB)^{†}.$

$\langle 21\rangle$ $((BC)^{†}A^{†})^{†}(BC)^{†}A^{†}ABC = ABC$ and $ABCC^{†}(AB)^{†}(C^{†}(AB)^{†})^{†} = ABC.$

$\langle 22\rangle$ $ABC(ABC)^{†} = ((BC)^{†}A^{†})^{†}(BC)^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}(AB)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 23\rangle$ $(A^{†}ABCC^{†})^{†} = (ABCC^{†})^{†}A$ and $(A^{†}ABCC^{†})^{†} = C(A^{†}ABC)^{†}.$

$\langle 24\rangle$ $(ABCC^{†})^{†} = (A^{†}ABCC^{†})^{†}A^{†}$ and $(A^{†}ABC)^{†} = C^{†}(A^{†}ABCC^{†})^{†}.$

$\langle 25\rangle$ $(A^{†}ABC)^{†}A^{†} = C^{†}(ABCC^{†})^{†}.$

$\langle 26\rangle$ $(ABCC^{†})^{†}A = C(A^{†}ABC)^{†}.$

$\langle 27\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{\ast}(A^{†}ABCC^{†})^{†}A^{\ast}.$

$\langle 28\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast} = (A^{†}ABCC^{†})^{†}.$

$\langle 29\rangle$ $(A^{†})^{\ast}((C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast})^{†}(C^{†})^{\ast} = (A^{†})^{\ast}B(C^{†})^{\ast}.$

$\langle 30\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}((C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast})^{†}(C^{†})^{\ast})^{†}(A^{†})^{\ast} = (C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 31\rangle$ $A^{†}ABCC^{†} = A^{\ast}(C^{\ast}(A^{†}ABCC^{†})^{†}A^{\ast})^{†}C^{\ast}.$

$\langle 32 \rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{\ast}((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast}B(C^{†})^{\ast} (A^{†}AB(C^{†})^{\ast})^{†}A^{\ast}.$

$\langle 33\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast} = ((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast}B(C^{†})^{\ast} (A^{†}AB(C^{†})^{\ast})^{†}.$

$\langle 34\rangle$ $((A^{†})^{\ast}BC)^{†} = (A^{†}ABC)^{†}A^{\ast}$ and $(AB(C^{†})^{\ast})^{†} = C^{\ast}(ABCC^{†})^{†}.$

$\langle 35\rangle$ $(A^{†}ABC)^{†} = ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}$ and $(ABCC^{†})^{†} = (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 36\rangle$ $ABC(ABC)^{†} = AB(C^{†})^{\ast}(A^{†}AB(C^{†})^{\ast})^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast}BC.$

$\langle 37\rangle$ $ABC(ABC)^{†}A = AB(C^{†})^{\ast}(A^{†}AB(C^{†})^{\ast})^{†}$ and $C(ABC)^{†}ABC = ((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast}BC.$

$\langle 38\rangle$ $ABC(ABC)^{†} = ((B(C^{†})^{\ast})^{†}A^{†})^{†}(B(C^{†})^{\ast})^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}((A^{†})^{\ast}B)^{†}(C^{†}((A^{†})^{\ast}B)^{†})^{†}.$

$\langle 39\rangle$ $(BC)^{†}A^{†}ABC((A^{†})^{\ast}BC)^{†} = (BC)^{†}A^{\ast}$ and $(AB(C^{†})^{\ast})^{†}ABCC^{†}(AB)^{†} = C^{\ast}(AB)^{†}.$

$\langle 40\rangle$ $((BC)^{†}A^{\ast})^{†}(BC)^{†}A^{†}ABC = (A^{†})^{\ast}BC$ and $ABCC^{†}(AB)^{†}(C^{\ast}(AB)^{†})^{†} = AB(C^{†})^{\ast}.$

$\langle 41\rangle$ $(A^{†}ABCC^{†})^{†} = ((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast}$ and $(A^{†}ABCC^{†})^{†} = (C^{†})^{\ast}(A^{†}AB(C^{†})^{\ast})^{†}.$

$\langle 42\rangle$ $((A^{†})^{\ast}BCC^{†})^{†} = (A^{†}ABCC^{†})^{†}A^{\ast}$ and $(A^{†}AB(C^{†})^{\ast})^{†} = C^{\ast}(A^{†}ABCC^{†})^{†}.$

$\langle 43\rangle$ $(A^{†}AB(C^{†})^{\ast})^{†}A^{\ast} = C^{\ast}((A^{†})^{\ast}BCC^{†})^{†}.$

$\langle 44\rangle$ $((A^{†})^{\ast}BCC^{†})^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(A^{†}AB(C^{†})^{\ast})^{†}.$

$\langle 45\rangle$ $C^{\ast}((A^{†})^{\ast}BCC^{†})^{†} = (A^{†}AB(C^{†})^{\ast})^{†}A^{\ast}.$

$\langle 46\rangle$ $(ABC)^{†} = C^{\ast}(A^{\ast}ABCC^{\ast})^{†}A^{\ast}.$

$\langle 47\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = (C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast}.$

$\langle 48\rangle$ $(ABC)^{†} = C^{\ast}(ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}A^{\ast}.$

$\langle 49\rangle$ $(C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast} = (ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}.$

$\langle 50\rangle$ $(ABC)^{†} = (A^{\ast}ABC)^{†}A^{\ast}$ and $(ABC)^{†} = C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 51\rangle$ $(A^{\ast}ABC)^{†} = (ABC)^{†}(A^{†})^{\ast}$ and $(ABCC^{\ast})^{†} = (C^{†})^{\ast}(ABC)^{†}.$

$\langle 52\rangle$ $((A^{†})^{\ast}BC)^{†} = ((A^{\ast}A)^{†}BC)^{†}A^{†}$ and $(AB(C^{†})^{\ast})^{†} = C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 53\rangle$ $((A^{\ast}A)^{†}BC)^{†} = ((A^{†})^{\ast}BC)^{†}A$ and $(AB(CC^{\ast})^{†})^{†} = C(AB(C^{†})^{\ast})^{†}.$

$\langle 54\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = ((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{†}$ and $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{†}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 55\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = ((A^{†})^{\ast}B(C^{†})^{\ast})^{†}A$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = C((A^{†})^{\ast}B(C^{†})^{\ast})^{†}.$

$\langle 56\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{†}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{†}.$

$\langle 57\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = C((A^{†})^{\ast}B(C^{†})^{\ast})^{†}A.$

$\langle 58\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = (ABCC^{\ast})^{†}(A^{†})^{\ast}$ and $(A^{\ast}ABCC^{\ast})^{†} = (C^{†})^{\ast}(A^{\ast}ABC)^{†}.$

$\langle 59\rangle$ $(ABCC^{\ast})^{†} = (A^{\ast}ABCC^{\ast})^{†}A^{\ast}$ and $(A^{\ast}ABC)^{†} = C^{\ast}(A^{\ast}ABCC^{\ast})^{†}.$

$\langle 60\rangle$ $(A^{\ast}ABC)^{†}A^{\ast} = C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 61\rangle$ $(ABCC^{\ast})^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(A^{\ast}ABC)^{†}.$

$\langle 62\rangle$ $(AB(CC^{\ast})^{†})^{†} = (A^{\ast}AB(CC^{\ast})^{†})^{†}A^{\ast}$ and $((A^{\ast}A)^{†}BC)^{†} = C^{\ast}((A^{\ast}A)^{†}BCC^{\ast})^{†}.$

$\langle 63\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (AB(CC^{\ast})^{†})^{†}(A^{†})^{\ast}$ and $((A^{\ast}A)^{†}BCC^{\ast})^{†} = (C^{†})^{\ast}((A^{\ast}A)^{†}BC)^{†}.$

$\langle 64\rangle$ $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}A = C((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}.$

$\langle 65\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{†} = C^{†}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 66\rangle$ $(A^{\ast}ABC)^{†} = (AA^{\ast}ABC)^{†}A$ and $(ABCC^{\ast})^{†} = C(ABCC^{\ast}C)^{†}.$

$\langle 67\rangle$ $(AA^{\ast}ABC)^{†} = (A^{\ast}ABC)^{†}A^{†}$ and $(ABCC^{\ast}C)^{†} = C^{†}(ABCC^{\ast})^{†}.$

$\langle 68\rangle$ $((A^{\ast}A)^{†}BC)^{†} = ((A^{\ast}AA^{\ast})^{†}BC)^{†}(A^{†})^{\ast}$ and $(AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(AB(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 69\rangle$ $((A^{\ast}A)^{†}BC)^{†}A^{\ast} = ((A^{\ast}AA^{\ast})^{†}BC)^{†}$ and $C^{\ast}(AB(CC^{\ast})^{†})^{†} = (AB(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 70\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = ((A^{\ast}AA^{\ast})^{†}B(C^{†})^{\ast})^{†}(A^{†})^{\ast}$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{†})^{\ast}B(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 71\rangle$ $((A^{\ast}AA^{\ast})^{†}B(C^{†})^{\ast})^{†} = ((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{\ast}$ and $((A^{†})^{\ast}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 72\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = C(A^{\ast}ABCC^{\ast}C)^{†}$ and $(A^{\ast}ABCC^{\ast})^{†} = (AA^{\ast}ABCC^{\ast})^{†}A.$

$\langle 73\rangle$ $(A^{\ast}ABCC^{\ast}C)^{†} = C^{†}(A^{\ast}ABCC^{\ast})^{†}$ and $(AA^{\ast}ABCC^{\ast})^{†} = (A^{\ast}ABCC^{\ast})^{†}A^{†}.$

$\langle 74\rangle$ $((A^{\ast}A)^{†}BCC^{\ast})^{†} = C((A^{\ast}A)^{†}BCC^{\ast}C)^{†}$ and $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}A.$

$\langle 75\rangle$ $((A^{\ast}A)^{†}BCC^{\ast}C)^{†} = C^{†}((A^{\ast}A)^{†}BCC^{\ast})^{†}$ and $(A^{\ast}AB(CC^{\ast})^{†})^{†}A^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}.$

$\langle 76\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}$ and $((A^{\ast}A)^{†}BCC^{\ast})^{†} = ((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 77\rangle$ $(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}(A^{\ast}AB(CC^{\ast})^{†})^{†}$ and $(AA^{\ast}AB(CC^{\ast})^{†})^{†} = (A^{\ast}AB(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 78\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{\ast}A)^{†}B(C^{\ast}CC^{\ast})^{†})^{†}$ and

$((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = ((A^{\ast}AA^{\ast})^{†}B(CC^{\ast})^{†})^{†}(A^{†})^{\ast}.$

$\langle 79\rangle$ $((A^{\ast}A)^{†}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}$ and $((A^{\ast}AA^{\ast})^{†}B(CC^{\ast})^{†})^{†} = ((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 80\rangle$ $C(A^{\ast}ABCC^{\ast}C)^{†} = (AA^{\ast}ABCC^{\ast})^{†}A.$

$\langle 81\rangle$ $(A^{\ast}ABCC^{\ast}C)^{†}A^{†} = C^{†}(AA^{\ast}ABCC^{\ast})^{†}.$

$\langle 82\rangle$ $C((A^{\ast}A)^{†}BCC^{\ast}C)^{†} = ((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 83\rangle$ $((A^{\ast}A)^{†}BCC^{\ast}C)^{†}A^{\ast} = C^{†}((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}.$

$\langle 84\rangle$ $(C^{†})^{\ast}(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}A.$

$\langle 85\rangle$ $(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}A^{†} = C^{\ast}(AA^{\ast}AB(CC^{\ast})^{†})^{†}.$

$\langle 86\rangle$ $((CC^{\ast}C)^{†}B^{\ast}(A^{\ast}A)^{†})^{†}C^{†} = A^{†}((CC^{\ast})^{†}B^{\ast}(AA^{\ast}A)^{†})^{†}.$

$\langle 87\rangle$ $A((CC^{\ast}C)^{†}B^{\ast}(A^{\ast}A)^{†})^{†} = ((CC^{\ast})^{†}B^{\ast}(AA^{\ast}A)^{†})^{†}C.$

$\langle 88\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = C(AA^{\ast}ABCC^{\ast}C)^{†}A.$

$\langle 89\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{†}(A^{\ast}ABCC^{\ast})^{†}A^{†}.$

$\langle 90\rangle$ $((A^{\ast}A)^{†}BCC^{\ast})^{†} = C((A^{\ast}AA^{\ast})^{†} BCC^{\ast}C)^{†}(A^{†})^{\ast}.$

$\langle 91\rangle$ $((A^{\ast}AA^{\ast})^{†} BCC^{\ast}C)^{†} = C^{†}((A^{\ast}A)^{†}BCC^{\ast})^{†}A^{\ast}.$

$\langle 92\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(AA^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}A.$

$\langle 93\rangle$ $(AA^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}(A^{\ast}AB(CC^{\ast})^{†})^{†} (A^{†})^{\ast}.$

$\langle 94\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{\ast}AA^{\ast})^{†}B(C^{\ast}CC^{\ast})^{†})^{†}(A^{†})^{\ast}.$

$\langle 95\rangle$ $((A^{\ast}AA^{\ast})^{†}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 96\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = ((A^{\ast}A)^2BCC^{\ast}C)^{†}A^{\ast}$ and $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{\ast}(AA^{\ast}AB(CC^{\ast})^2)^{†}.$

$\langle 97\rangle$ $((A^{\ast}A)^2BCC^{\ast}C)^{†} = (AA^{\ast}ABCC^{\ast}C)^{†}(A^{†})^{\ast}$ and $(AA^{\ast}AB(CC^{\ast})^2)^{†} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†}.$

$\langle 98\rangle$ $((A^{\ast}A)^2BCC^{\ast}C)^{†}A^{\ast} = C^{\ast}(AA^{\ast}AB(CC^{\ast})^2)^{†}.$

$\langle 99\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†}.$

$\langle 100\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{\ast}((A^{\ast}A)^2B(CC^{\ast})^2)^{†}A^{\ast}.$

$\langle 101\rangle$ $((A^{\ast}A)^2B(CC^{\ast})^2)^{†} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†} (A^{†})^{\ast}.$

$\langle 102\rangle$ $(CC^{†}B^{\ast}A^{†}A)^{†} = (C^{†}B^{\ast}A^{†}A)^{†}C^{†}$ and $(CC^{†}B^{\ast}A^{†}A)^{†} = A^{†}(CC^{†}B^{\ast}A^{†})^{†}.$

$\langle 103\rangle$ $(C^{†}B^{\ast}A^{†}A)^{†} = (CC^{†}B^{\ast}A^{†}A)^{†}C$ and $(CC^{†}B^{\ast}A^{†})^{†} = A(CC^{†}B^{\ast}A^{†}A)^{†}.$

$\langle 104\rangle$ $(A^{\ast}ABCC^{†})^{†} = C(A^{\ast}ABC)^{†}$ and $(A^{†}ABCC^{\ast})^{†} = (ABCC^{\ast})^{†}A.$

$\langle 105\rangle$ $(A^{\ast}ABC)^{†} = C^{†}(A^{\ast}ABCC^{†})^{†}$ and $(ABCC^{\ast})^{†} = (A^{†}ABCC^{\ast})^{†}A^{†}.$

$\langle 106\rangle$ $(C^{†}B^{\ast}(A^{\ast}A)^{†})^{†} = ((CC^{\ast})^{†}B^{\ast}(A^{\ast}A)^{†})^{†}(C^{†})^{\ast}$ and $((CC^{\ast})^{†}B^{\ast}A^{†})^{†} = (A^{†})^{\ast}((CC^{\ast})^{†}B^{\ast}(A^{\ast}A)^{†})^{†}.$

$\langle 107\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = C^{†}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = ((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} A^{†}.$

$\langle 108\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = C((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}$ and $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = ((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}A.$

$\langle 109\rangle$ $(ABC(BC)^{†}A^{†})^{†} = ((BC)^{†}A^{†})^{†}(ABC)^{†}$ and $(C^{†}(AB)^{†}ABC)^{†} = (ABC)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 110\rangle$ $ABC(A^{†}ABC)^{†}A^{†}$ and $C^{†}(ABCC^{†})^{†}ABC$ are orthogonal projectors$.$

$\langle 111\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ and $C^{\ast}(ABCC^{\ast})^{†}ABC$ are orthogonal projectors$.$

$\langle 112\rangle$ $A^{†}(ABC)(ABC)^{†}A$ and $C(ABC)^{†}(ABC)C^{†}$ orthogonal projectors$.$

$\langle 113\rangle$ $ABC(ABC)^{†}$ and $AA^{\ast}$ commute$,$ and $(ABC)^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 114\rangle$ $ABC(A^{†}ABC)^{†}A^{†}$ and $AA^{\ast}$ commute$,$ and $C^{†}(ABCC^{†})^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 115\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ and $AA^{\ast}$ commute$,$ and $C^{\ast}(ABCC^{\ast})^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 116\rangle$ $A^{†}(ABC)(ABC)^{†}A$ and $A^{\ast}A$ commute$,$ and $C(ABC)^{†}(ABC)C^{†}$ and $CC^{\ast}$ commute$.$

$\langle 117\rangle$ $A((BC)^{†}A^{†}A)^{†}(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}(CC^{†}(AB)^{†})^{†}C$ are orthogonal projectors$.$

$\langle 118\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ are orthogonal projectors$.$

$\langle 119\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ and $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ are orthogonal projectors$.$

$\langle 120\rangle$ $((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})$ and $(AA^{\ast})^{†}$ commute$,$ and $(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 121\rangle$ $A((BC)^{†}A^{†}A)^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$,$ and $C^{†}(AB)^{†}(CC^{†}(AB)^{†})^{†}C$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 122\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$,$ and $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 123\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ and $(A^{\ast}A)^{†}$ commute$,$ and $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ and $(CC^{\ast})^{†}$ commute$.$

$\langle 124\rangle$ $ABC(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}ABC$ are EP$.$

$\langle 125\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (A^{†}ABC)^{†}A^{\ast}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}(ABCC^{†})^{†}.$

$\langle 126\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni ((A^{\ast}A)^{†}BC)^{†}A^{†}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 127\rangle$ $\{(A^{†}ABC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}$ and $\{(ABCC^{†})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 128\rangle$ $\{((A^{\ast}A)^{†}BC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}A$ and $\{(AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}.$

$\langle 129\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{†}(A^{†}ABC(BC)^{†})^{†}A^{\ast}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{†}ABCC^{†})^{†}(AB)^{†}.$

$\langle 130\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{\ast}((A^{\ast}A)^{†}BC(BC)^{\ast})^{†}A^{†}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(A^{\ast}AB(CC^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 131\rangle$ $\{(A^{†}ABC(BC)^{†})^{(1, 2, 3)}\} \ni BC((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}$ and $\{((AB)^{†}ABCC^{†})^{(1, 2, 4)}\} \ni(C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}AB.$

$\langle 132\rangle$ $\{((A^{\ast}A)^{†}BC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}((A^{†})^{\ast}BC)^{†}A$

and $\{((AB)^{\ast}AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}((AB)^{†})^{\ast}.$

$\langle 133\rangle$ $(AA^{\ast}ABC)(AA^{\ast}ABC)^{†} = (ABC)(ABC)^{†}$ and $(ABCC^{\ast}C)^{†}(ABCC^{\ast}C) = (ABC)^{†}(ABC).$

$\langle 134\rangle$ ${\mathscr R}((ABC)^{†}) = {\mathscr R}(C^{†}(A^{†}ABCC^{†})^{†}A^{†})$ and ${\mathscr R}(((ABC)^{†})^{\ast}) = {\mathscr R}((C^{†}(A^{†}ABCC^{†})^{†}A^{†})^{\ast}).$

$\langle 135\rangle$ ${\mathscr R}(AA^{\ast}ABC) = {\mathscr R}(ABC)$ and ${\mathscr R}((ABCC^{\ast}C)^{\ast}) = {\mathscr R}((ABC)^{\ast}).$

$\langle 136\rangle$ $r[AA^{\ast}ABC, \, ABC] = r[(ABCC^{\ast}C)^{\ast}, \, (ABC)^{\ast}] = r(ABC).$

$\langle 137\rangle$ $r[(A^{\ast}AA^{\ast})^{†}BC, \, (A^{†})^{\ast}BC] = r((A^{†})^{\ast}BC)$ and $r[(CC^{\ast}C)^{†}(AB)^{\ast}, \, C^{†}(AB)^{\ast}] = r(C^{†}(AB)^{\ast}).$

Proof. It is easy to verify by the definition of Moore–Penrose generalized inverse and (3.9) that the two facts $(ABC)^{†} \in \{(ABC)^{(2)} \}$ and $C^{†}(A^{†}ABCC^{†})^{†}A^{†} \in \{(ABC)^{(2)}\}$ hold. In this case, applying (2.12) to $(ABC)^{†}$ and $C^{†}(A^{†}ABCC^{†})^{†}A^{†}$ leads to the equivalence of $\langle 1\rangle$ and $\langle 134\rangle$. Furthermore, applying (2.11) to the difference $(ABC)^{†} - C^{†}(A^{†}ABCC^{†})^{†}A^{†}$ and simplifying by (2.13) and Lemma 2.7, we obtain the following rank equalities

Setting all sides of (3.11) equal to zero and noticing that $r\begin{bmatrix} ABC \\ ABCC^{\ast}C \end{bmatrix} \geq r(ABC)$ and $r[ABC, \, AA^{\ast}ABC] \geq r(ABC)$ leads to

$r\begin{bmatrix} ABC \\ ABCC^{\ast}C \end{bmatrix} = r[ABC, \, AA^{\ast}ABC] = r(AA^{\ast}ABC) = r(ABCC^{\ast}C) = r(ABC)$, thus establishing the equivalences of $\langle 1\rangle$, $\langle 135\rangle$, and $\langle 136\rangle$ by Lemma 2.2.

The equivalence of $\langle 136\rangle$ and $\langle 137\rangle$ follows from Theorem 3.2 $\langle 61\rangle$ and $\langle 62\rangle$ and Theorem 3.3 $\langle 61\rangle$ and $\langle 62\rangle$.

Pre- and post-multiplying the equality in $\langle 1\rangle$ with $C$ and $A$, respectively, yield the equality in $\langle 2\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 2\rangle$ with $C^{†}$ and $A^{†}$, respectively, and simplifying yield the equality in $\langle 1\rangle$.

We first take the Moore–Penrose generalized inverses of both sides of the equality in $\langle 2\rangle$ to yield $(C(ABC)^{†}A)^{†} = A^{†}ABCC^{†}$. We then pre- and post-multiplying the equality with $A$ and $C$, respectively, to obtain the equality in $\langle 3\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 3\rangle$ with $A^{†}$ and $C^{†}$, respectively, and simplifying yield $(C(ABC)^{†}A)^{†} = A^{†}ABCC^{†}$, which is equivalent to $\langle 2\rangle$ by taking the Moore–Penrose generalized inverses of both sides of the equality.

Result $\langle 3\rangle$ obviously implies $\langle 4\rangle$ by substitution. Conversely, pre- and post-multiplying the equality in $\langle 4\rangle$ with $C^{†}$ and $A^{†}$, respectively, and simplifying yield the equality in $\langle 3\rangle$.

Taking the Moore–Penrose generalized inverses of both sides of the equality in $\langle 1\rangle$, and then pre- and post-multiplying the equality with $A^{†}$ and $C^{†}$, respectively, we obtain the equality in $\langle 5\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 5\rangle$ with $A$ and $C$, respectively, and then taking the Moore–Penrose generalized inverses of both sides of the equality, we obtain $\langle 1\rangle$.

Notice by (3.10) that $C^{†}(ABCC^{†})^{†}ABC(A^{†}ABC)^{†}A^{†} \in \{(ABC)^{(2)} \}$ holds. Then applying (2.11) to the difference $(ABC)^{†} - C^{†}(ABCC^{†})^{†}ABC(A^{†}ABC)^{†}A^{†}$ and simplifying by (2.13) and Lemma 2.7, we obtain

Setting all sides of (3.12) equal to zero leads to the equivalence of $\langle 6\rangle$ and $\langle 136\rangle$.

Pre- and post-multiplying the equality in $\langle 6\rangle$ with $A$ and $C$, respectively, and simplifying we obtain the equality in $\langle 7\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 7\rangle$ with $A^{†}$ and $C^{†}$, respectively, we obtain $\langle 6\rangle$.

Combining Theorem 3.2 $\langle 1\rangle$–$\langle 8\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 1\rangle$–$\langle 8\rangle$ and $\langle 60\rangle$ leads to the equivalences of $\langle 8\rangle$–$\langle 16\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 13\rangle$–$\langle 18\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 13\rangle$–$\langle 18\rangle$ and $\langle 60\rangle$ leads to the equivalences of $\langle 17\rangle$–$\langle 22\rangle$ and $\langle 136\rangle$.

It is easy to verify that $(ABCC^{†})^{†}A \in \{(A^{†}ABCC^{†})^{(1, 2)} \}$ and $C(A^{†}ABC)^{†} \in \{(A^{†}ABCC^{†})^{(1, 2)} \}$ hold. In this case, applying (2.11) to the two differences $(A^{†}ABCC^{†})^{†} - (ABCC^{†})^{†}A$ and $(A^{†}ABCC^{†})^{†} - C(A^{†}ABC)^{†}$ and simplifying by (2.13) and Lemma 2.7 yield

and

Setting all sides of (3.13) and (3.14) equal to zero leads to the equivalence of $\langle 23\rangle$ and $\langle 136\rangle$.

Post- and pre-multiplying the two equalities in $\langle 20\rangle$ with $A^{†}$ and $C^{†}$, respectively, and simplifying yield the two equalities in $\langle 24\rangle$. Conversely, post- and pre-multiplying the two equalities in $\langle 24\rangle$ with $A$ and $C$, respectively, and simplifying yield the two equalities in $\langle 23\rangle$.

Notice by (3.7) and (3.8) that $(A^{†}ABC)^{†}A^{†} \in \{(ABC)^{(2)} \}$ and $C^{†}(ABCC^{†})^{†} \in \{(ABC)^{(2)} \}$ hold. Hence applying (2.11) to $(A^{†}ABC)^{†}A^{†} - C^{†}(ABCC^{†})^{†}$ and simplifying by (2.13) and Lemma 2.7 yield

Setting all sides of (3.15) equal to zero leads to the equivalence of $\langle 25\rangle$ and $\langle 136\rangle$.

Post- and pre-multiplying the equality in $\langle 25\rangle$ with $A$ and $C$, respectively, yield the equality in $\langle 26\rangle$. Conversely, post- and pre-multiplying the equality in $\langle 26\rangle$ with $A^{†}$ and $C^{†}$, respectively, and simplifying yield the equality in $\langle 25\rangle$.

Replacing $A$ and $C$ by $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$, respectively, in $\langle 1\rangle$–$\langle 7\rangle$ and $\langle 16\rangle$–$\langle 26\rangle$ leads to the equivalences of these results and $\langle 27\rangle$–$\langle 45\rangle$ through $\langle 136\rangle$ and $\langle 137\rangle$.

It is easy to verify by the definition of Moore–Penrose generalized inverse that $C^{\ast}(A^{\ast}ABCC^{\ast})^{†}A^{\ast} \in \{(ABC)^{(2)} \}$ holds. In this case, applying (2.11) to the difference $(ABC)^{†} - C^{\ast}(A^{\ast}ABCC^{\ast})^{†}A^{\ast}$ and simplifying by (2.13) and Lemma 2.7, we obtain

Setting all sides of (3.16) equal to zero leads to the equivalence of $\langle 46\rangle$ and $\langle 136\rangle$.

Pre- and post-multiplying the equality in $\langle 46\rangle$ with $(C^{†})^{\ast}$ and $(A^{†})^{\ast}$, respectively, and simplifying yield the equality in $\langle 47\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 47\rangle$ with $C^{†}$ and $A^{†}$, respectively, and simplifying yield the equality in $\langle 46\rangle$.

It is easy to verify by the definition of Moore–Penrose generalized inverse that $C^{\ast}(ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}A^{\ast} \in \{(ABC)^{(2)} \}$ holds. Then applying (2.11) to the difference $(ABC)^{†} - C^{\ast}(ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}A^{\ast}$ and simplifying by (2.13) and Lemma 2.7, we obtain

Setting all sides of (3.17) equal to zero leads to the equivalence of $\langle 48\rangle$ and $\langle 136\rangle$.

Pre- and post-multiplying the equality in $\langle 48\rangle$ with $(C^{†})^{\ast}$ and $(A^{†})^{\ast}$, respectively, and simplifying yield the equality in $\langle 49\rangle$. Conversely, pre- and post-multiplying the equality in $\langle 49\rangle$ with $C^{†}$ and $A^{†}$, respectively, and simplifying yield the equality in $\langle 48\rangle$.

The equivalence of $\langle 1\rangle$ and $\langle 50\rangle$ follows from combining Theorem 3.2 $\langle 1\rangle$ and $\langle 19\rangle$ and Theorem 3.3 $\langle 1\rangle$ and $\langle 19\rangle$.

Post- and pre-multiplying the equality in $\langle 50\rangle$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$, respectively, and simplifying yield the equality in $\langle 51\rangle$. Conversely, post- and pre-multiplying the equality in $\langle 51\rangle$ with $^{\ast}$ and $C^{\ast}$, respectively, and simplifying yield the equality in $\langle 50\rangle$.

Replacing $A$ and $C$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$ in $\langle 50\rangle$ and $\langle 51\rangle$, respectively, and simplifying yield the equivalences $\langle 1\rangle$ with $\langle 52\rangle$ and $\langle 53\rangle$ through equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

Replacing $A$ and $C$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$ in $\langle 50\rangle$ and $\langle 51\rangle$, simultaneously, and simplifying yield the equivalences $\langle 1\rangle$ with $\langle 54\rangle$ and $\langle 55\rangle$ through equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

Replacing $A$ and $C$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$ in $\langle 46\rangle$ and $\langle 47\rangle$, simultaneously, and simplifying yield the equivalences $\langle 1\rangle$ with $\langle 56\rangle$ and $\langle 57\rangle$ through equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

It is easy to verify that $(ABCC^{\ast})^{†}(A^{†})^{\ast} \in \{(A^{\ast}ABCC^{\ast})^{(2)} \}$ and $(C^{†})^{\ast}(A^{\ast}ABC)^{†} \in \{(A^{\ast}ABCC^{\ast})^{(2)} \}$ hold. In this case, applying (2.11) to the difference $(A^{\ast}ABCC^{\ast})^{†} - (ABCC^{\ast})^{†}(A^{†})^{\ast}$ and $(A^{\ast}ABCC^{\ast})^{†} - (C^{†})^{\ast}(A^{\ast}ABC)^{†}$ and simplifying by (2.13) and Lemma 2.7 yields

and

Setting all sides of (3.18) and (3.19) equal to zero leads to the equivalence of $\langle 58\rangle$ and $\langle 136\rangle$.

Post- and pre-multiplying the equality in $\langle 58\rangle$ with $A^{\ast}$ and $C^{\ast}$, respectively, and simplifying yield the equality in $\langle 59\rangle$. Conversely, post- and pre-multiplying the equality in $\langle 58\rangle$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$, respectively, and simplifying yield the equality in $\langle 59\rangle$.

It is easy to verify that $(A^{\ast}ABC)^{†}A^{\ast} \in \{(ABC)^{(2)} \}$ and $C^{\ast}(ABCC^{\ast})^{†} \in \{(ABC)^{(2)} \}$ hold. Then applying (2.11), (2.13), and Lemma 2.7,

Setting all sides of (3.20) equal to zero leads to the equivalence of $\langle 60\rangle$ and $\langle 136\rangle$.

Pre- and post-multiplying the equality in $\langle 60\rangle$ with $(C^{†})^{\ast}$ and $(A^{†})^{\ast}$, respectively, and simplifying yield the equality in $\langle 61\rangle$. Conversely, post- and pre-multiplying the equality in $\langle 61\rangle$ with $C^{\ast}$ and $A^{\ast}$, respectively, and simplifying yield the equality in $\langle 61\rangle$.

Replacing $C$ and $A$ with $(C^{†})^{\ast}$ and $(A^{†})^{\ast}$ in the two equalities in $\langle 58\rangle$, respectively, and simplifying yield the equivalence of $\langle 1\rangle$ with $\langle 62\rangle$ through the equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

Post- and pre-multiplying the two equalities in $\langle 62\rangle$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$, respectively, and simplifying yield the two equalities in $\langle 63\rangle$. Conversely, post- and pre-multiplying the equality in $\langle 61\rangle$ with $A^{\ast}$ and $C^{\ast}$, respectively, and simplifying yield the equality in $\langle 62\rangle$.

Replacing $A$ and $C$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$ in the equality in $\langle 61\rangle$, respectively, and simplifying yield the equivalence $\langle 1\rangle$ with $\langle 64\rangle$ through the equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

Replacing $A$ and $C$ with $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$ in the equality in $\langle 60\rangle$, respectively, and simplifying yield the equivalence $\langle 1\rangle$ with $\langle 65\rangle$ through the equivalence of $\langle 136\rangle$ and $\langle 137\rangle$.

The derivations of (3.11)–(3.20) present typical steps of establishing and simplifying matrix rank equalities associated with nested ROLs in the corresponding statements. The equivalences of $\langle 66\rangle$–$\langle 108\rangle$ with $\langle 136\rangle$ and $\langle 137\rangle$ can be established by similar approaches, and the routine verifications are left to the reader.

Combining Theorem 3.2 $\langle 26\rangle$ and $\langle 60\rangle$ and Theorem 3.3 $\langle 26\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalence of $\langle 109\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 27\rangle$–$\langle 29\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 27\rangle$–$\langle 29\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalences of $\langle 110\rangle$–$\langle 112\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 32\rangle$–$\langle 35\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 32\rangle$–$\langle 35\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalences of $\langle 113\rangle$–$\langle 116\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 49\rangle$–$\langle 51\rangle$ and $\langle 60\rangle$ and Theorem 3.3 $\langle 49\rangle$–$\langle 51\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalences of $\langle 117\rangle$–$\langle 119\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 54\rangle$–$\langle 57\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 54\rangle$–$\langle 57\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalences of $\langle 120\rangle$–$\langle 123\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 36\rangle$ and $\langle 60\rangle$ and Theorem 3.3 $\langle 36\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalence of $\langle 124\rangle$ and $\langle 136\rangle$.

Combining Theorem 3.2 $\langle 37\rangle$–$\langle 44\rangle$ and $\langle 60\rangle$ with Theorem 3.3 $\langle 37\rangle$–$\langle 44\rangle$ and $\langle 60\rangle$, and comparing them with $\langle 136\rangle$, we obtain the equivalences of $\langle 125\rangle$–$\langle 132\rangle$ and $\langle 136\rangle$.

The equivalence of $\langle 133\rangle$ and $\langle 135\rangle$ follows from Lemma 2.2 (b).

With a bit more work we can also obtain the following result.

Theorem 3.5. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 3 statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = C^{†}(A^{†}ABCC^{†})^{†}A^{†}.$

$\langle 2\rangle$ $(ABC)^{†} = C^{†}(A^{†}AB + BCC^{†} - B)^{†}A^{†}.$

$\langle 3\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{\ast}(A^{†}AB + BCC^{†} - B)^{†}A^{\ast}.$

Proof. Since $A^{†}A$ and $CC^{†}$ are orthogonal projectors, we obtain by Lemma 2.9 the following matrix identity

Substituting it into the equality in $\langle 1\rangle$ and simplifying lead to the equivalence of $\langle 1\rangle$ and $\langle 2\rangle$. Replacing $A$ and $C$ in $\langle 1\rangle$ by $(A^{†})^{\ast}$ and $(C^{†})^{\ast}$, respectively, yields $\langle 3\rangle$.

Finally, we present a pair of equivalent facts associated with (1.12).

Corollary 3.6. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 2 statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = C^{†}(A^{†}ABCC^{†})^{†}A^{†}.$

$\langle 2\rangle$ $((A^{\ast}A)^{1/2}B(CC^{\ast})^{1/2})^{†} = ((CC^{\ast})^{1/2})^{†}(A^{†}ABCC^{†})^{†}((A^{\ast}A)^{1/2})^{†}.$

Proof. It is well recognized in matrix theory that the two products $A^{\ast}A$ and $CC^{\ast}$ are positive semi-definite matrices, and the two square roots $(A^{\ast}A)^{1/2}$ and $(CC^{\ast})^{1/2}$ make sense and are unique. In this situation,

hold, so that

Hold. Now applying Theorem 3.4 $\langle 1\rangle$ and $\langle 136\rangle$ to the matrix product $(A^{\ast}A)^{1/2}B(CC^{\ast})^{1/2}$, we obtain

where by Lemmas 2.4 and 2.7,

Combining the above two groups of equalities with Theorem 3.4 $\langle 1\rangle$ and $\langle 136\rangle$ leads to the equivalence of $\langle 1\rangle$ and $\langle 2\rangle$ in the theorem.

Various consequences can be derived form Theorems 3.2–3.4. For example, (1.12) is reduced to the second equality in (1.3) if $A^{†}ABCC^{†} = B$, which includes $A^{†}A = I_n$ and $CC^{†} = I_p$ as its special case. Thus we obtain the following results from Theorem 3.4.

Corollary 3.7. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q},$ and assume that $A^{†}ABCC^{†} = B$ holds$,$ i.e.$,$ ${\mathscr R}(B) \subseteq {\mathscr R}(A^{\ast})$ and ${\mathscr R}(B^{\ast}) \subseteq {\mathscr R}(C).$ Then the following 137 statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = C^{†}B^{†}A^{†}.$

$\langle 2\rangle$ $C(ABC)^{†}A = B^{†}.$

$\langle 3\rangle$ $A(C(ABC)^{†}A)^{†}C = ABC.$

$\langle 4\rangle$ $C(A(C(ABC)^{†}A)^{†}C)^{†}A = C(ABC)^{†}A.$

$\langle 5\rangle$ $B = A^{†}(C^{†}B^{†}A^{†})^{†}C^{†}.$

$\langle 6 \rangle$ $(ABC)^{†} = C^{†}(AB)^{†}ABC(BC)^{†}A^{†}.$

$\langle 7\rangle$ $C(ABC)^{†}A = (AB)^{†}ABC(BC)^{†}.$

$\langle 8\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{†}A^{†}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}(AB)^{†}.$

$\langle 9\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (A^{\ast}ABC)^{†}A^{\ast}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 10\rangle$ $\{(BC)^{(1, 2, 3)}\} \ni (ABC)^{†}A$ and $\{(AB)^{(1, 2, 4)}\} \ni C(ABC)^{†}.$

$\langle 11\rangle$ $\{(A^{\ast}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}(A^{†})^{\ast}$ and $\{(ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}.$

$\langle 12\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{†}(BC(BC)^{†})^{†}A^{†}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{†}((AB)^{†}AB)^{†}(AB)^{†}.$

$\langle 13\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (BC)^{\ast}(A^{\ast}ABC(BC)^{\ast})^{†}A^{\ast}$ and $\{(ABC)^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{\ast}ABCC^{\ast})^{†}(AB)^{\ast}.$

$\langle 14\rangle$ $\{(BC(BC)^{†})^{(1, 2, 3)}\} \ni BC(ABC)^{†}A$ and $\{((AB)^{†}AB)^{(1, 2, 4)}\} \ni C(ABC)^{†}AB.$

$\langle 15\rangle$ $\{(A^{\ast}ABC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast}$ and $\{(A^{\ast}ABCC^{\ast})^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast}.$

$\langle 16\rangle$ $(ABC)^{†} = (BC)^{†}A^{†}$ and $(ABC)^{†} = C^{†}(AB)^{†}.$

$\langle 17\rangle$ $(BC)^{†} = (ABC)^{†}A$ and $(AB)^{†} = C(ABC)^{†}.$

$\langle 18\rangle$ $ABC(ABC)^{†} = ABC(BC)^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}(AB)^{†}ABC.$

$\langle 19\rangle$ $ABC(ABC)^{†}A = ABC(BC)^{†}$ and $C(ABC)^{†}ABC = (AB)^{†}ABC.$

$\langle 20\rangle$ $(BC)^{†}BC(ABC)^{†} = (BC)^{†}A^{†}$ and $(ABC)^{†}AB(AB)^{†} = C^{†}(AB)^{†}.$

$\langle 21\rangle$ $((BC)^{†}A^{†})^{†}(BC)^{†}BC = ABC$ and $AB(AB)^{†}(C^{†}(AB)^{†})^{†} = ABC.$

$\langle 22\rangle$ $ABC(ABC)^{†} = ((BC)^{†}A^{†})^{†}(BC)^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}(AB)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 23\rangle$ $B^{†} = (AB)^{†}A$ and $B^{†} = C(BC)^{†}.$

$\langle 24\rangle$ $(AB)^{†} = B^{†}A^{†}$ and $(BC)^{†} = C^{†}B^{†}.$

$\langle 25\rangle$ $(BC)^{†}A^{†} = C^{†}(AB)^{†}.$

$\langle 26\rangle$ $(AB)^{†}A = C(BC)^{†}.$

$\langle 27\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{\ast}B^{†}A^{\ast}.$

$\langle 28\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast} = B^{†}.$

$\langle 29\rangle$ $(A^{†})^{\ast}((C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast})^{†}(C^{†})^{\ast} = (A^{†})^{\ast}B(C^{†})^{\ast}.$

$\langle 30\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}((C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast})^{†}(C^{†})^{\ast})^{†}(A^{†})^{\ast} = (C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 31\rangle$ $B = A^{\ast}(C^{\ast}B^{†}A^{\ast})^{†}C^{\ast}.$

$\langle 32 \rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{\ast}((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}B(C^{†})^{\ast} (B(C^{†})^{\ast})^{†}A^{\ast}.$

$\langle 33\rangle$ $(C^{†})^{\ast}((A^{†})^{\ast}B(C^{†})^{\ast})^{†}(A^{†})^{\ast} = ((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}B(C^{†})^{\ast} (B(C^{†})^{\ast})^{†}.$

$\langle 34\rangle$ $((A^{†})^{\ast}BC)^{†} = (BC)^{†}A^{\ast}$ and $(AB(C^{†})^{\ast})^{†} = C^{\ast}(AB)^{†}.$

$\langle 35\rangle$ $(BC)^{†} = ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}$ and $(AB)^{†} = (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 36\rangle$ $ABC(ABC)^{†} = AB(C^{†})^{\ast}(B(C^{†})^{\ast})^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}BC.$

$\langle 37\rangle$ $ABC(ABC)^{†}A = AB(C^{†})^{\ast}(B(C^{†})^{\ast})^{†}$ and $C(ABC)^{†}ABC = ((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}BC.$

$\langle 38\rangle$ $ABC(ABC)^{†} = ((B(C^{†})^{\ast})^{†}A^{†})^{†}(B(C^{†})^{\ast})^{†}A^{†}$ and $(ABC)^{†}ABC = C^{†}((A^{†})^{\ast}B)^{†}(C^{†}((A^{†})^{\ast}B)^{†})^{†}.$

$\langle 39\rangle$ $(BC)^{†}BC((A^{†})^{\ast}BC)^{†} = (BC)^{†}A^{\ast}$ and $(AB(C^{†})^{\ast})^{†}AB(AB)^{†} = C^{\ast}(AB)^{†}.$

$\langle 40\rangle$ $((BC)^{†}A^{\ast})^{†}(BC)^{†}BC = (A^{†})^{\ast}BC$ and $AB(AB)^{†}(C^{\ast}(AB)^{†})^{†} = AB(C^{†})^{\ast}.$

$\langle 41\rangle$ $B^{†} = ((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}$ and $B^{†} = (C^{†})^{\ast}(B(C^{†})^{\ast})^{†}.$

$\langle 42\rangle$ $((A^{†})^{\ast}B)^{†} = B^{†}A^{\ast}$ and $(B(C^{†})^{\ast})^{†} = C^{\ast}B^{†}.$

$\langle 43\rangle$ $(B(C^{†})^{\ast})^{†}A^{\ast} = C^{\ast}((A^{†})^{\ast}B)^{†}.$

$\langle 44\rangle$ $((A^{†})^{\ast}B)^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(B(C^{†})^{\ast})^{†}.$

$\langle 45\rangle$ $C^{\ast}((A^{†})^{\ast}B)^{†} = (B(C^{†})^{\ast})^{†}A^{\ast}.$

$\langle 46\rangle$ $(ABC)^{†} = C^{\ast}(A^{\ast}ABCC^{\ast})^{†}A^{\ast}.$

$\langle 47\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = (C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast}.$

$\langle 48\rangle$ $(ABC)^{†} = C^{\ast}(ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}A^{\ast}.$

$\langle 49\rangle$ $(C^{†})^{\ast}(ABC)^{†}(A^{†})^{\ast} = (ABCC^{\ast})^{†}ABC(A^{\ast}ABC)^{†}.$

$\langle 50\rangle$ $(ABC)^{†} = (A^{\ast}ABC)^{†}A^{\ast}$ and $(ABC)^{†} = C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 51\rangle$ $(A^{\ast}ABC)^{†} = (ABC)^{†}(A^{†})^{\ast}$ and $(ABCC^{\ast})^{†} = (C^{†})^{\ast}(ABC)^{†}.$

$\langle 52\rangle$ $((A^{†})^{\ast}BC)^{†} = ((A^{\ast}A)^{†}BC)^{†}A^{†}$ and $(AB(C^{†})^{\ast})^{†} = C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 53\rangle$ $((A^{\ast}A)^{†}BC)^{†} = ((A^{†})^{\ast}BC)^{†}A$ and $(AB(CC^{\ast})^{†})^{†} = C(AB(C^{†})^{\ast})^{†}.$

$\langle 54\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = ((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{†}$ and $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{†}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 55\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = ((A^{†})^{\ast}B(C^{†})^{\ast})^{†}A$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = C((A^{†})^{\ast}B(C^{†})^{\ast})^{†}.$

$\langle 56\rangle$ $((A^{†})^{\ast}B(C^{†})^{\ast})^{†} = C^{†}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{†}.$

$\langle 57\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = C((A^{†})^{\ast}B(C^{†})^{\ast})^{†}A.$

$\langle 58\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = (ABCC^{\ast})^{†}(A^{†})^{\ast}$ and $(A^{\ast}ABCC^{\ast})^{†} = (C^{†})^{\ast}(A^{\ast}ABC)^{†}.$

$\langle 59\rangle$ $(ABCC^{\ast})^{†} = (A^{\ast}ABCC^{\ast})^{†}A^{\ast}$ and $(A^{\ast}ABC)^{†} = C^{\ast}(A^{\ast}ABCC^{\ast})^{†}.$

$\langle 60\rangle$ $(A^{\ast}ABC)^{†}A^{\ast} = C^{\ast}(ABCC^{\ast})^{†}.$

$\langle 61\rangle$ $(ABCC^{\ast})^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(A^{\ast}ABC)^{†}.$

$\langle 62\rangle$ $(AB(CC^{\ast})^{†})^{†} = (A^{\ast}AB(CC^{\ast})^{†})^{†}A^{\ast}$ and $((A^{\ast}A)^{†}BC)^{†} = C^{\ast}((A^{\ast}A)^{†}BCC^{\ast})^{†}.$

$\langle 63\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (AB(CC^{\ast})^{†})^{†}(A^{†})^{\ast}$ and $((A^{\ast}A)^{†}BCC^{\ast})^{†} = (C^{†})^{\ast}((A^{\ast}A)^{†}BC)^{†}.$

$\langle 64\rangle$ $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}A = C((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}.$

$\langle 65\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{†} = C^{†}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 66\rangle$ $(A^{\ast}ABC)^{†} = (AA^{\ast}ABC)^{†}A$ and $(ABCC^{\ast})^{†} = C(ABCC^{\ast}C)^{†}.$

$\langle 67\rangle$ $(AA^{\ast}ABC)^{†} = (A^{\ast}ABC)^{†}A^{†}$ and $(ABCC^{\ast}C)^{†} = C^{†}(ABCC^{\ast})^{†}.$

$\langle 68\rangle$ $((A^{\ast}A)^{†}BC)^{†} = ((A^{\ast}AA^{\ast})^{†}BC)^{†}(A^{†})^{\ast}$ and $(AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(AB(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 69\rangle$ $((A^{\ast}A)^{†}BC)^{†}A^{\ast} = ((A^{\ast}AA^{\ast})^{†}BC)^{†}$ and $C^{\ast}(AB(CC^{\ast})^{†})^{†} = (AB(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 70\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = ((A^{\ast}AA^{\ast})^{†}B(C^{†})^{\ast})^{†}(A^{†})^{\ast}$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{†})^{\ast}B(C^{\ast}CC^{\ast})^{†})^{†}.$

$\langle 71\rangle$ $((A^{\ast}AA^{\ast})^{†}B(C^{†})^{\ast})^{†} = ((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}A^{\ast}$ and $((A^{†})^{\ast}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}.$

$\langle 72\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = C(A^{\ast}ABCC^{\ast}C)^{†}$ and $(A^{\ast}ABCC^{\ast})^{†} = (AA^{\ast}ABCC^{\ast})^{†}A.$

$\langle 73\rangle$ $(A^{\ast}ABCC^{\ast}C)^{†} = C^{†}(A^{\ast}ABCC^{\ast})^{†}$ and $(AA^{\ast}ABCC^{\ast})^{†} = (A^{\ast}ABCC^{\ast})^{†}A^{†}.$

$\langle 74\rangle$ $((A^{\ast}A)^{†}BCC^{\ast})^{†} = C((A^{\ast}A)^{†}BCC^{\ast}C)^{†}$ and $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}A.$

$\langle 75\rangle$ $((A^{\ast}A)^{†}BCC^{\ast}C)^{†} = C^{†}((A^{\ast}A)^{†}BCC^{\ast})^{†}$ and $(A^{\ast}AB(CC^{\ast})^{†})^{†}A^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}.$

$\langle 76\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}$ and $((A^{\ast}A)^{†}BCC^{\ast})^{†} = ((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 77\rangle$ $(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}(A^{\ast}AB(CC^{\ast})^{†})^{†}$ and $(AA^{\ast}AB(CC^{\ast})^{†})^{†} = (A^{\ast}AB(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 78\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{\ast}A)^{†}B(C^{\ast}CC^{\ast})^{†})^{†}$

and $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = ((A^{\ast}AA^{\ast})^{†}B(CC^{\ast})^{†})^{†}(A^{†})^{\ast}.$

$\langle 79\rangle$ $((A^{\ast}A)^{†}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}$ and $((A^{\ast}AA^{\ast})^{†}B(CC^{\ast})^{†})^{†} = ((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 80\rangle$ $C(A^{\ast}ABCC^{\ast}C)^{†} = (AA^{\ast}ABCC^{\ast})^{†}A.$

$\langle 81\rangle$ $(A^{\ast}ABCC^{\ast}C)^{†}A^{†} = C^{†}(AA^{\ast}ABCC^{\ast})^{†}.$

$\langle 82\rangle$ $C((A^{\ast}A)^{†}BCC^{\ast}C)^{†} = ((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}(A^{†})^{\ast}.$

$\langle 83\rangle$ $((A^{\ast}A)^{†}BCC^{\ast}C)^{†}A^{\ast} = C^{†}((A^{\ast}AA^{\ast})^{†}BCC^{\ast})^{†}.$

$\langle 84\rangle$ $(C^{†})^{\ast}(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = (AA^{\ast}AB(CC^{\ast})^{†})^{†}A.$

$\langle 85\rangle$ $(A^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}A^{†} = C^{\ast}(AA^{\ast}AB(CC^{\ast})^{†})^{†}.$

$\langle 86\rangle$ $((CC^{\ast}C)^{†}B^{\ast}(A^{\ast}A)^{†})^{†}C^{†} = A^{†}((CC^{\ast})^{†}B^{\ast}(AA^{\ast}A)^{†})^{†}.$

$\langle 87\rangle$ $A((CC^{\ast}C)^{†}B^{\ast}(A^{\ast}A)^{†})^{†} = ((CC^{\ast})^{†}B^{\ast}(AA^{\ast}A)^{†})^{†}C.$

$\langle 88\rangle$ $(A^{\ast}ABCC^{\ast})^{†} = C(AA^{\ast}ABCC^{\ast}C)^{†}A.$

$\langle 89\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{†}(A^{\ast}ABCC^{\ast})^{†}A^{†}.$

$\langle 90\rangle$ $((A^{\ast}A)^{†}BCC^{\ast})^{†} = C((A^{\ast}AA^{\ast})^{†} BCC^{\ast}C)^{†}(A^{†})^{\ast}.$

$\langle 91\rangle$ $((A^{\ast}AA^{\ast})^{†} BCC^{\ast}C)^{†} = C^{†}((A^{\ast}A)^{†}BCC^{\ast})^{†}A^{\ast}.$

$\langle 92\rangle$ $(A^{\ast}AB(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}(AA^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†}A.$

$\langle 93\rangle$ $(AA^{\ast}AB(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}(A^{\ast}AB(CC^{\ast})^{†})^{†} (A^{†})^{\ast}.$

$\langle 94\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = (C^{†})^{\ast}((A^{\ast}AA^{\ast})^{†}B(C^{\ast}CC^{\ast})^{†})^{†}(A^{†})^{\ast}.$

$\langle 95\rangle$ $((A^{\ast}AA^{\ast})^{†}B(C^{\ast}CC^{\ast})^{†})^{†} = C^{\ast}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}A^{\ast}.$

$\langle 96\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = ((A^{\ast}A)^2BCC^{\ast}C)^{†}A^{\ast}$ and $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{\ast}(AA^{\ast}AB(CC^{\ast})^2)^{†}.$

$\langle 97\rangle$ $((A^{\ast}A)^2BCC^{\ast}C)^{†} = (AA^{\ast}ABCC^{\ast}C)^{†}(A^{†})^{\ast}$ and $(AA^{\ast}AB(CC^{\ast})^2)^{†} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†}.$

$\langle 98\rangle$ $((A^{\ast}A)^2BCC^{\ast}C)^{†}A^{\ast} = C^{\ast}((AA^{\ast}AB(CC^{\ast})^2)^{†}.$

$\langle 99\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†}(A^{†})^{\ast} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†}.$

$\langle 100\rangle$ $(AA^{\ast}ABCC^{\ast}C)^{†} = C^{\ast}((A^{\ast}A)^2B(CC^{\ast})^2)^{†}A^{\ast}.$

$\langle 101\rangle$ $((A^{\ast}A)^2B(CC^{\ast})^2)^{†} = (C^{†})^{\ast}(AA^{\ast}ABCC^{\ast}C)^{†} (A^{†})^{\ast}.$

$\langle 102\rangle$ $(B^{\ast})^{†} = A^{†}(B^{\ast}A^{†})^{†}$ and $(B^{\ast})^{†} = (C^{†}B^{\ast})^{†}C^{†}.$

$\langle 103\rangle$ $(B^{\ast}A^{†})^{†} = A(B^{\ast})^{†}$ and $(C^{†}B^{\ast})^{†} = (B^{\ast})^{†}C.$

$\langle 104\rangle$ $(A^{\ast}AB)^{†} = C(A^{\ast}ABC)^{†}$ and $(BCC^{\ast})^{†} = (ABCC^{\ast})^{†}A.$

$\langle 105\rangle$ $(A^{\ast}ABC)^{†} = C^{†}(A^{\ast}AB)^{†}$ and $(ABCC^{\ast})^{†} = (BCC^{\ast})^{†}A^{†}.$

$\langle 106\rangle$ $(C^{†}B^{\ast}(A^{\ast}A)^{†})^{†} = ((CC^{\ast})^{†}B^{\ast}(A^{\ast}A)^{†})^{†}(C^{†})^{\ast}$ and $((CC^{\ast})^{†}B^{\ast}A^{†})^{†} = (A^{†})^{\ast}((CC^{\ast})^{†}B^{\ast}(A^{\ast}A)^{†})^{†}.$

$\langle 107\rangle$ $((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†} = C^{†}((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†}$ and $((A^{†})^{\ast}B(CC^{\ast})^{†})^{†} = ((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} A^{†}.$

$\langle 108\rangle$ $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = C((A^{\ast}A)^{†}B(C^{†})^{\ast})^{†}$ and $((A^{\ast}A)^{†}B(CC^{\ast})^{†})^{†} = ((A^{†})^{\ast}B(CC^{\ast})^{†})^{†}A.$

$\langle 109\rangle$ $(ABC(BC)^{†}A^{†})^{†} = ((BC)^{†}A^{†})^{†}(ABC)^{†}$ and $(C^{†}(AB)^{†}ABC)^{†} = (ABC)^{†}(C^{†}(AB)^{†})^{†}.$

$\langle 110\rangle$ $ABC(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}ABC$ are orthogonal projectors$.$

$\langle 111\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ and $C^{\ast}(ABCC^{\ast})^{†}ABC$ are orthogonal projectors$.$

$\langle 112\rangle$ $A^{†}(ABC)(ABC)^{†}A$ and $C(ABC)^{†}(ABC)C^{†}$ orthogonal projectors$.$

$\langle 113\rangle$ $ABC(ABC)^{†}$ and $AA^{\ast}$ commute$,$ and $(ABC)^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 114\rangle$ $ABC(BC)^{†}A^{†}$ and $AA^{\ast}$ commute$,$ and $C^{†}(AB)^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 115\rangle$ $ABC(A^{\ast}ABC)^{†}A^{\ast}$ and $AA^{\ast}$ commute$,$ and $C^{\ast}(ABCC^{\ast})^{†}ABC$ and $C^{\ast}C$ commute$.$

$\langle 116\rangle$ $A^{†}(ABC)(ABC)^{†}A$ and $A^{\ast}A$ commute$,$ and $C(ABC)^{†}(ABC)C^{†}$ and $CC^{\ast}$ commute$.$

$\langle 117\rangle$ $ABC(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}ABC$ are orthogonal projectors$.$

$\langle 118\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ are orthogonal projectors$.$

$\langle 119\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ and $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ are orthogonal projectors$.$

$\langle 120\rangle$ $((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})$ and $(AA^{\ast})^{†}$ commute$,$ and $(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 121\rangle$ $A((BC)^{†})^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$,$ and $C^{†}(AB)^{†}((AB)^{†})^{†}C$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 122\rangle$ $(A^{†})^{\ast}((BC)^{†}(A^{\ast}A)^{†})^{†}(BC)^{†}A^{†}$ and $(AA^{\ast})^{†}$ commute$,$ and $C^{†}(AB)^{†}((CC^{\ast})^{†}(AB)^{†})^{†}(C^{†})^{\ast}$ and $(C^{\ast}C)^{†}$ commute$.$

$\langle 123\rangle$ $A^{†}((BC)^{†}A^{†})^{†}((BC)^{†}A^{†})A$ and $(A^{\ast}A)^{†}$ commute$,$ and $C(C^{†}(AB)^{†})(C^{†}(AB)^{†})^{†}C^{†}$ and $(CC^{\ast})^{†}$ commute$.$

$\langle 124\rangle$ $ABC(BC)^{†}A^{†}$ and $C^{†}(AB)^{†}ABC$ are EP$.$

$\langle 125\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{†}A^{\ast}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}(AB)^{†}.$

$\langle 126\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni ((A^{\ast}A)^{†}BC)^{†}A^{†}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(AB(CC^{\ast})^{†})^{†}.$

$\langle 127\rangle$ $\{(BC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}(A^{†})^{\ast}$ and $\{(AB)^{(1, 2, 4)}\} \ni (C^{†})^{\ast}(AB(C^{†})^{\ast})^{†}.$

$\langle 128\rangle$ $\{((A^{\ast}A)^{†}BC)^{(1, 2, 3)}\} \ni ((A^{†})^{\ast}BC)^{†}A$ and $\{(AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}.$

$\langle 129\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{†}(BC(BC)^{†})^{†}A^{\ast}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{\ast}((AB)^{†}AB)^{†}(AB)^{†}.$

$\langle 130\rangle$ $\{((A^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{\ast}((A^{\ast}A)^{†}BC(BC)^{\ast})^{†}A^{†}$ and $\{(AB(C^{†})^{\ast})^{(1, 2, 4)}\} \ni C^{†}(A^{\ast}AB(CC^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 131\rangle$ $\{(BB^{†})^{(1, 2, 3)}\} \ni B((A^{†})^{\ast}B)^{†}(A^{†})^{\ast}$ and $\{(B^{†}B)^{(1, 2, 4)}\} \ni(C^{†})^{\ast}(B(C^{†})^{\ast})^{†}B.$

$\langle 132\rangle$ $\{((A^{\ast}A)^{†}BC(BC)^{\ast})^{(1, 2, 3)}\} \ni ((BC)^{†})^{\ast}((A^{†})^{\ast}BC)^{†}A$

and $\{((AB)^{\ast}AB(CC^{\ast})^{†})^{(1, 2, 4)}\} \ni C(AB(C^{†})^{\ast})^{†}((AB)^{†})^{\ast}.$

$\langle 133\rangle$ $(A^{\ast}AB)(A^{\ast}AB)^{†} = BB^{†}$ and $(BCC^{\ast})^{†}(BCC^{\ast}) = B^{†}B.$

$\langle 134\rangle$ ${\mathscr R}(((AB)^{†})^{\ast}) = {\mathscr R}((B^{†}A^{†})^{\ast})$ and ${\mathscr R}((BC)^{†}) = {\mathscr R}(C^{†}B^{†}).$

$\langle 135\rangle$ ${\mathscr R}(A^{\ast}AB) = {\mathscr R}(B)$ and ${\mathscr R}((BCC^{\ast})^{\ast}) = {\mathscr R}(B^{\ast}).$

$\langle 136\rangle$ $r[A^{\ast}AB, \, B] = r[(BCC^{\ast})^{\ast}, \, B^{\ast}] = r(B).$

$\langle 137\rangle$ $r[(A^{\ast}AA^{\ast})^{†}B, \, (A^{†})^{\ast}B] = r(B)$ and $r[(CC^{\ast}C)^{†}B^{\ast}, \, C^{†}B^{\ast}] = r(B).$

4.   Main results II

The analysis carried out above for the nested ROLs in (1.10)–(1.15) can similarly be done for the nested ROLs in (1.16)–(1.21) by comparing (1.23) and (1.24) and making symbolic replacements of the results in Lemma 3.1, Theorems 3.2–3.5, and Corollary 3.6. The details are readily presented below without proofs.

Lemma 4.1. ^[26] Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$

$\langle 1\rangle$ The following 3 matrix set inclusions always hold

$\langle 2\rangle$ The following 3 matrix set inclusions always hold

$\langle 3\rangle$ The following 4 matrix set inclusions always hold

$\langle 4\rangle$ The two matrix equalities in (1.19) and (1.20) always hold$.$ In particular$,$ if $(AB)^{†}ABC = B^{†}BC$ and $ABC(BC)^{†} = ABB^{†},$ then the nested ROL $(ABC)^{†} = (B^{†}BC)^{†}B^{†}(ABB^{†})^{†}$ holds$.$

Theorem 4.2. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni ((AB)^{†}ABC)^{†}(AB)^{†}.$

$\langle 2\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni ((AB)^{\ast}ABC)^{†}(AB)^{\ast}.$

$\langle 3\rangle$ $\{((AB)^{†}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}AB.$

$\langle 4\rangle$ $\{((AB)^{\ast}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}((AB)^{†})^{\ast}.$

$\langle 5\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{†}((AB)^{†}ABC(B^{†}BC)^{†})^{†} (AB)^{†}.$

$\langle 6\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{\ast}((AB)^{\ast}ABC(B^{†}BC)^{\ast})^{†}(AB)^{\ast}.$

$\langle 7\rangle$ $\{((AB)^{†}ABC(B^{†}BC)^{†})^{(1, 2, 3)}\} \ni B^{†}BC(ABC)^{†}AB.$

$\langle 8\rangle$ $\{((AB)^{\ast}ABC(B^{†}BC)^{\ast})^{(1, 2, 3)}\} \ni ((B^{†}BC)^{†})^{\ast}(ABC)^{†} ((AB)^{†})^{\ast}.$

$\langle 9\rangle$ $(ABC)^{†} = ((AB)^{†}ABC)^{†}(AB)^{†}.$

$\langle 10\rangle$ $ABC = (((AB)^{†}ABC)^{†}(AB)^{†})^{†}.$

$\langle 11\rangle$ $ABC = AB((ABC)^{†}AB)^{†}.$

$\langle 12\rangle$ $(ABC)^{†}AB = ((AB)^{†}ABC)^{†}.$

$\langle 13\rangle$ $BC(ABC)^{†}AB = BC((AB)^{†}ABC)^{†}.$

$\langle 14\rangle$ $ABC(ABC)^{†} = ABC((AB)^{†}ABC)^{†}(AB)^{†}.$

$\langle 15\rangle$ $ABC(ABC)^{†}AB = ABC((AB)^{†}ABC)^{†}.$

$\langle 16\rangle$ $(B^{†}BC)^{†}(AB)^{†}ABC(ABC)^{†} = (B^{†}BC)^{†}(AB)^{†}.$

$\langle 17\rangle$ $((B^{†}BC)^{†}(AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}ABC = ABC.$

$\langle 18\rangle$ $ABC(ABC)^{†} = ((B^{†}BC)^{†}(AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}.$

$\langle 19\rangle$ $(ABC)^{†} = ((AB)^{\ast}ABC)^{†}(AB)^{\ast}.$

$\langle 20\rangle$ $ABC = (((AB)^{\ast}ABC)^{†}(AB)^{\ast})^{†}.$

$\langle 21\rangle$ $ABC = ((AB)^{†})^{\ast}((ABC)^{†}((AB)^{†})^{\ast})^{†}.$

$\langle 22\rangle$ $(ABC)^{†}((AB)^{†})^{\ast} = ((AB)^{\ast}ABC)^{†}.$

$\langle 23\rangle$ $BC(ABC)^{†}((AB)^{†})^{\ast} = BC((AB)^{\ast}ABC)^{†}.$

$\langle 24\rangle$ $ABC(ABC)^{†} = ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}.$

$\langle 25\rangle$ $ABC(ABC)^{†}((AB)^{†})^{\ast} = ABC((AB)^{\ast}ABC)^{†}.$

$\langle 26\rangle$ $(ABC(B^{†}BC)^{†}(AB)^{†})^{†} = ((B^{†}BC)^{†}(AB)^{†})^{†}(ABC)^{†}.$

$\langle 27\rangle$ $ABC((AB)^{†}ABC)^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 28\rangle$ $ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ is an orthogonal projector$.$

$\langle 29\rangle$ $(AB)^{†}(ABC)(ABC)^{†}AB$ is an orthogonal projector$.$

$\langle 30\rangle$ $AB((AB)^{†}ABC(B^{†}BC)^{†})^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 31\rangle$ $ABC(B^{†}BC)^{\ast}((AB)^{\ast}ABC(B^{†}BC)^{\ast})^{†}(AB)^{\ast}$ is an orthogonal projector$.$

$\langle 32\rangle$ $ABC(ABC)^{†}$ and $AB(AB)^{\ast}$ commute$.$

$\langle 33\rangle$ $ABC((AB)^{†}ABC)^{†}(AB)^{†}$ and $AB(AB)^{\ast}$ commute$.$

$\langle 34\rangle$ $ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ and $AB(AB)^{\ast}$ commute$.$

$\langle 35\rangle$ $(AB)^{†}(ABC)(ABC)^{†}AB$ and $(AB)^{\ast}AB$ commute$.$

$\langle 36\rangle$ $ABC(B^{†}BC)^{†}(AB)^{†}$ is EP$.$

$\langle 37\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\} \ni ((AB)^{†}ABC)^{†}(AB)^{\ast}.$

$\langle 38\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\} \ni(((AB)^{\ast}AB)^{†}B^{†}BC)^{†}(AB)^{†}.$

$\langle 39\rangle$ $\{((AB)^{†}ABC)^{(1, 2, 3)}\} \ni (((AB)^{†})^{\ast}B^{†}BC)^{†}((AB)^{†})^{\ast}.$

$\langle 40\rangle$ $\{(((AB)^{\ast}AB)^{†}B^{†}BC)^{(1, 2, 3)}\} \ni (((AB)^{†})^{\ast}B^{†}BC)^{†}AB.$

$\langle 41\rangle$ $\{(((ABB^{†})^{†})^{\ast}BC)^{(1, 2, 3)}\} \ni (BC)^{†}((ABB^{†})^{†}ABC(BC)^{†})^{†}(ABB^{†})^{\ast}.$

$\langle 42\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{\ast}(((AB)^{\ast}AB)^{†}B^{†}BC(B^{†}BC)^{\ast})^{†}(AB)^{†}.$

$\langle 43\rangle$ $\{((AB)^{†}ABC(B^{†}BC)^{†})^{(1, 2, 3)}\} \ni B^{†}BC(((AB)^{†})^{\ast}BC)^{†}((AB)^{†})^{\ast}.$

$\langle 44\rangle$ $\{(((AB)^{\ast}AB)^{†}B^{†}BC(B^{†}BC)^{\ast})^{(1, 2, 3)}\} \ni ((B^{†}BC)^{†})^{\ast}(((AB)^{†})^{\ast}B^{†}BC)^{†}AB.$

$\langle 45\rangle$ $(((AB)^{†})^{\ast}B^{†}BC)^{†} = ((AB)^{†}ABC)^{†}(AB)^{\ast}.$

$\langle 46\rangle$ $(((AB)^{†})^{\ast}B^{†}BC)^{†} = (((AB)^{\ast}AB)^{†}B^{†}BC)^{†}(AB)^{†}.$

$\langle 47\rangle$ $((AB)^{†}ABC)^{†} = (((AB)^{†})^{\ast}B^{†}BC)^{†}((AB)^{†})^{\ast}.$

$\langle 48\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC)^{†} = (((AB)^{†})^{\ast}B^{†}BC)^{†}AB.$

$\langle 49\rangle$ $AB((B^{†}BC)^{†}(AB)^{†}AB)^{†}(B^{†}BC)^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 50\rangle$ $((AB)^{†})^{\ast}((B^{†}BC)^{†}((AB)^{\ast}AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 51\rangle$ $(AB)^{†}((B^{†}BC)^{†}(AB)^{†})^{†}((B^{†}BC)^{†}(AB)^{†})AB$ is an orthogonal projector$.$

$\langle 52\rangle$ $AB(B^{†}BC(B^{†}BC)^{†}(AB)^{†}AB)^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 53\rangle$ $((AB)^{†})^{\ast}(B^{†}BC(B^{†}BC)^{\ast})^{†}(((AB)^{\ast}AB)^{†}(B^{†}BC(B^{†}BC)^{\ast})^{†})^{†}(AB)^{†}$ is an orthogonal projector$.$

$\langle 54\rangle$ $((B^{†}BC)^{†}(AB)^{†})^{†}((B^{†}BC)^{†}(AB)^{†})$ and $(AB(AB)^{\ast})^{†}$ commute$.$

$\langle 55\rangle$ $AB((B^{†}BC)^{†}(AB)^{†}(AB))^{†}(B^{†}BC)^{†}(AB)^{†}$ and $(AB(AB)^{\ast})^{†}$ commute$.$

$\langle 56\rangle$ $((AB)^{†})^{\ast}((B^{†}BC)^{†}((AB)^{\ast}AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}$ and $((AB)(AB)^{\ast})^{†}$ commute$.$

$\langle 57\rangle$ $(AB)^{†}((B^{†}BC)^{†}(AB)^{†})^{†}((B^{†}BC)^{†}(AB)^{†})AB$ and $((AB)^{\ast}AB)^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}((AB)(AB)^{\ast}ABC) = {\mathscr R}(ABC).$

$\langle 59\rangle$ ${\mathscr R}((AB)^{†})^{\ast}B^{†}BC) = {\mathscr R}(ABC).$

$\langle 60\rangle$ $r[(AB)(AB)^{\ast}ABC, \, ABC] = r(ABC).$

$\langle 61\rangle$ $r[((AB)^{†})^{\ast}B^{†}BC, \, ABC] = r(ABC).$

$\langle 62\rangle$ $r[((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}BC, \, ((AB)^{†})^{\ast}B^{†}BC] = r(((AB)^{†})^{\ast}B^{†}BC).$

Theorem 4.3. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 62 statements are equivalent$:$

$\langle 1\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{†}(ABC(BC)^{†})^{†}.$

$\langle 2\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{\ast}(ABC(BC)^{\ast})^{†}.$

$\langle 3\rangle$ $\{(ABC(BC)^{†})^{(1, 2, 4)}\} \ni BC(ABC)^{†}.$

$\langle 4\rangle$ $\{(ABC(BC)^{\ast})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}.$

$\langle 5\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{†}((ABB^{†})^{†}ABC(BC)^{†})^{†}(ABB^{†})^{†}.$

$\langle 6\rangle$ $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{\ast}((ABB^{†})^{\ast}ABC(BC)^{\ast})^{†}(ABB^{†})^{\ast}.$

$\langle 7\rangle$ $\{((ABB^{†})^{†}ABC(BC)^{†})^{(1, 2, 4)}\} \ni BC(ABC)^{†}ABB^{†}.$

$\langle 8\rangle$ $\{(ABB^{†})^{\ast}ABC(BC)^{\ast})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}((ABB^{†})^{†})^{\ast}.$

$\langle 9\rangle$ $(ABC)^{†} = (BC)^{†}(ABC(BC)^{†})^{†}.$

$\langle 10\rangle$ $ABC = ((BC)^{†}(ABC(BC)^{†})^{†})^{†}.$

$\langle 11\rangle$ $ABC = (BC(ABC)^{†})^{†}BC.$

$\langle 12\rangle$ $BC(ABC)^{†} = (ABC(BC)^{†})^{†}.$

$\langle 13\rangle$ $BC(ABC)^{†}AB = (ABC(BC)^{†})^{†}AB.$

$\langle 14\rangle$ $(ABC)^{†}ABC = (BC)^{†}(ABC(BC)^{†})^{†}ABC.$

$\langle 15\rangle$ $BC(ABC)^{†}ABC = (ABC(BC)^{†})^{†}ABC.$

$\langle 16\rangle$ $(ABC)^{†}ABC(BC)^{†}(ABB^{†})^{†} = (BC)^{†}(ABB^{†})^{†}.$

$\langle 17\rangle$ $ABC(BC)^{†}(ABB^{†})^{†}((BC)^{†}(ABB^{†})^{†})^{†} = ABC.$

$\langle 18\rangle$ $(ABC)^{†}ABC = (BC)^{†}(ABB^{†})^{†}((BC)^{†}(ABB^{†})^{†})^{†}.$

$\langle 19\rangle$ $(ABC)^{†} = (BC)^{\ast}(ABC(BC)^{\ast})^{†}.$

$\langle 20\rangle$ $ABC = ((BC)^{\ast}(ABC(BC)^{\ast})^{†})^{†}.$

$\langle 21\rangle$ $ABC = (((BC)^{†})^{\ast}(ABC)^{†})^{†}((BC)^{†})^{\ast}.$

$\langle 22\rangle$ $((BC)^{†})^{\ast}(ABC)^{†} = (ABC(BC)^{\ast})^{†}.$

$\langle 23\rangle$ $((BC)^{†})^{\ast}(ABC)^{†}AB = (ABC(BC)^{\ast})^{†}AB.$

$\langle 24\rangle$ $(ABC)^{†}ABC = (BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC.$

$\langle 25\rangle$ $((BC)^{†})^{\ast}(ABC)^{†}ABC = (ABC(BC)^{\ast})^{†}ABC.$

$\langle 26\rangle$ $((BC)^{†}(ABB^{†})^{†}ABC)^{†} = (ABC)^{†}((BC)^{†}(ABB^{†})^{†})^{†}.$

$\langle 27\rangle$ $(BC)^{†}(ABC(BC)^{†})^{†}ABC$ is an orthogonal projector$.$

$\langle 28\rangle$ $(BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC$ is an orthogonal projector$.$

$\langle 29\rangle$ $BC(ABC)^{†}(ABC)(BC)^{†}$ is an orthogonal projector$.$

$\langle 30\rangle$ $(BC)^{†}((ABB^{†})^{†}ABC(BC)^{†})^{†}BC$ is an orthogonal projector$.$

$\langle 31\rangle$ $(BC)^{\ast}((ABB^{†})^{\ast}ABC(BC)^{\ast})^{†}(ABB^{†})^{\ast}ABC$ is an orthogonal projector$.$

$\langle 32\rangle$ $(ABC)^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 33\rangle$ $(BC)^{†}(ABC(BC)^{†})^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 34\rangle$ $(BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 35\rangle$ $BC(ABC)^{†}(ABC)(BC)^{†}$ and $BC(BC)^{\ast}$ commute$.$

$\langle 36\rangle$ $(BC)^{†}(ABB^{†})^{†}ABC$ is EP$.$

$\langle 37\rangle$ $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni (BC)^{\ast}(ABC(BC)^{†})^{†}.$

$\langle 38\rangle$ $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni (BC)^{†}(ABB^{†}(BC(BC)^{\ast})^{†})^{†}.$

$\langle 39\rangle$ $\{(ABC(BC)^{†})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 40\rangle$ $\{(ABB^{†}((BC)(BC)^{\ast})^{†})^{(1, 2, 4)}\} \ni BC(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 41\rangle$ $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni (BC)^{\ast}((ABB^{†})^{†}ABC(BC)^{†})^{†}(ABB^{†})^{†}.$

$\langle 42\rangle$ $\{((ABB^{†})^{\ast}ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni (BC)^{†}(ABB^{†}(BC(BC)^{\ast})^{†})^{†}(ABB^{†})^{\ast}.$

$\langle 43\rangle$ $\{((ABB^{†})^{†}ABC(BC)^{†})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABB^{†}((BC)^{†})^{\ast})^{†}ABB^{†}.$

$\langle 44\rangle$ $\{((ABB^{†})^{\ast}ABB^{†}((BC)(BC)^{\ast})^{†})^{(1, 2, 4)}\} \ni BC(ABB^{†}((BC)^{†})^{\ast})^{†}((ABB^{†})^{†})^{\ast}.$

$\langle 45\rangle$ $(ABB^{†}((BC)^{†})^{\ast})^{†} = (BC)^{\ast}(ABC(BC)^{†})^{†}.$

$\langle 46\rangle$ $(ABB^{†}((BC)^{†})^{\ast}) = (BC)^{†}(ABB^{†}((BC)(BC)^{\ast})^{†})^{†}.$

$\langle 47\rangle$ $(ABC(BC)^{†})^{†} = ((BC)^{†})^{\ast}(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 48\rangle$ $(ABB^{†}((BC)^{†})^{\ast})^{†} = BC(ABB^{†}((BC)(BC)^{\ast})^{†})^{†}.$

$\langle 49\rangle$ $(BC)^{†}(ABB^{†})^{†}((BC)(BC)^{†}(ABB^{†})^{†})^{†}BC$ is an orthogonal projector$.$

$\langle 50\rangle$ $(BC)^{†}(ABB^{†})^{†}(((BC)(BC)^{\ast})^{†}(ABB^{†})^{†})^{†}((BC)^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 51\rangle$ $BC((BC)^{†}(ABB^{†})^{†})((BC)^{†}(ABB^{†})^{†})^{†}(BC)^{†}$ is an orthogonal projector$.$

$\langle 52\rangle$ $(BC)^{†}((BC)(BC)^{†}(ABB^{†})^{†}ABB^{†})^{†}BC$ is an orthogonal projector$.$

$\langle 53\rangle$ $(BC)^{†}(((ABB^{†})^{\ast}ABB^{†})^{†}((BC)(BC)^{\ast})^{†})^{†}((ABB^{†})^{\ast}ABB^{†})^{†} ((BC)^{†})^{\ast}$ is an orthogonal projector$.$

$\langle 54\rangle$ $((BC)^{†}(ABB^{†})^{†})((BC)^{†}(ABB^{†})^{†})^{†}$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 55\rangle$ $(BC)^{†}(ABB^{†})^{†}((BC)(BC)^{†}(ABB^{†})^{†})^{†}BC$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 56\rangle$ $(BC)^{†}(ABB^{†})^{†}(((BC)(BC)^{\ast})^{†}(ABB^{†})^{†})^{†}((BC)^{†})^{\ast}$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 57\rangle$ $BC((BC)^{†}(ABB^{†})^{†})((BC)^{†}(ABB^{†})^{†})^{†}(BC)^{†}$ and $((BC)(BC)^{\ast})^{†}$ commute$.$

$\langle 58\rangle$ ${\mathscr R}((BC)^{\ast}BC(BC)^{\ast}A^{\ast}) = {\mathscr R}((BC)^{\ast}A^{\ast}).$

$\langle 59\rangle$ ${\mathscr R}((BC)^{†}(ABB^{†})^{†}) = {\mathscr R}((BC)^{\ast}(ABB^{†})^{\ast}).$

$\langle 60\rangle$ $r[(BC)^{\ast}BC(ABC)^{\ast}, \, (ABC)^{\ast}] = r(ABC).$

$\langle 61\rangle$ $r[(BC)^{†}(ABB^{†})^{†}, \, (ABC)^{\ast}] = r(ABC).$

$\langle 62\rangle$ $r[(BC(BC)^{\ast}BC)^{†}(ABB^{†})^{†}, \, (BC)^{†}(ABB^{†})^{†}] = r((BC)^{†}(ABB^{†})^{†}).$

Theorem 4.4. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following 137 statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = (BC)^{†}((AB)^{†}ABC(BC)^{†})^{†}(AB)^{†}.$

$\langle 2\rangle$ $BC(ABC)^{†}AB = ((AB)^{†}ABC(BC)^{†})^{†}.$

$\langle 3\rangle$ $AB(BC(ABC)^{†}AB)^{†}BC = ABC.$

$\langle 4\rangle$ $BC(AB(BC(ABC)^{†}AB)^{†}BC)^{†}AB = BC(ABC)^{†}AB.$

$\langle 5\rangle$ $(AB)^{†}ABC(BC)^{†} = (AB)^{†}((BC)^{†}((AB)^{†}ABC(BC)^{†})^{†}(AB)^{†})^{†}(BC)^{†}.$

$\langle 6 \rangle$ $(ABC)^{†} = (BC)^{†}(ABC(BC)^{†})^{†}ABC((AB)^{†}ABC)^{†}(AB)^{†}.$

$\langle 7\rangle$ $BC(ABC)^{†}AB = (ABC(BC)^{†})^{†}ABC((AB)^{†}ABC)^{†}.$

$\langle 8\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni ((AB)^{†}ABC)^{†}(AB)^{†}$ and $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{†}(ABC(BC)^{†})^{†}.$

$\langle 9\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni ((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ and $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{\ast}(ABC(BC)^{\ast})^{†}.$

$\langle 10\rangle$ $\{(AB)^{†}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}AB$ and $\{(ABC(BC)^{†})^{(1, 2, 4)}\} \ni BC(ABC)^{†}.$

$\langle 11\rangle$ $\{((AB)^{\ast}ABC)^{(1, 2, 3)}\} \ni (ABC)^{†}((AB)^{†})^{\ast}$ and $\{(ABC(BC)^{\ast})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}.$

$\langle 12\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{†}((AB)^{†}ABC(B^{†}BC)^{†})^{†}(AB)^{†}$

and $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{†}((ABB^{†})^{†}ABC(BC)^{†})^{†}(ABB^{†})^{†}.$

$\langle 13\rangle$ $\{(ABC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{\ast}((AB)^{\ast}ABC(B^{†}BC)^{\ast})^{†}(AB)^{\ast}$

and $\{(ABC)^{(1, 2, 4)}\} \ni (BC)^{\ast}((ABB^{†})^{\ast}ABC(BC)^{\ast})^{†}(ABB^{†})^{\ast}.$

$\langle 14\rangle$ $\{((AB)^{†}ABC(B^{†}BC)^{†})^{(1, 2, 3)}\} \ni B^{†}BC(ABC)^{†}AB$

and $\{((ABB^{†})^{†}ABC(BC)^{†})^{(1, 2, 4)}\} \ni BC(ABC)^{†}ABB^{†}.$

$\langle 15\rangle$ $\{((AB)^{\ast}ABC(B^{†}BC)^{\ast})^{(1, 2, 3)}\} \ni ((B^{†}BC)^{†})^{\ast}(ABC)^{†} ((AB)^{†})^{\ast}$

and $\{((AB)^{\ast}ABC(BC)^{\ast})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABC)^{†}((ABB^{†})^{†})^{\ast}.$

$\langle 16\rangle$ $(ABC)^{†} = ((AB)^{†}ABC)^{†}(AB)^{†}$ and $(ABC)^{†} = (BC)^{†}(ABC(BC)^{†})^{†}.$

$\langle 17\rangle$ $((AB)^{†}ABC)^{†} = (ABC)^{†}AB$ and $(ABC(BC)^{†})^{†} = BC(ABC)^{†}.$

$\langle 18\rangle$ $ABC(ABC)^{†} = ABC((AB)^{†}ABC)^{†}(AB)^{†}$ and $(ABC)^{†}ABC = (BC)^{†}(ABC(BC)^{†})^{†}ABC.$

$\langle 19\rangle$ $ABC(ABC)^{†}AB = ABC((AB)^{†}ABC)^{†}$ and $BC(ABC)^{†}ABC = (ABC(BC)^{†})^{†}ABC.$

$\langle 20\rangle$ $(B^{†}BC)^{†}(AB)^{†}ABC(ABC)^{†} = (B^{†}BC)^{†}(AB)^{†}$ and $(ABC)^{†}ABC(BC)^{†}(ABB^{†})^{†} = (BC)^{†}(ABB^{†})^{†}.$

$\langle 21\rangle$ $((B^{†}BC)^{†}(AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}ABC = ABC$ and $ABC(BC)^{†}(ABB^{†})^{†}((BC)^{†}(ABB^{†})^{†})^{†} = ABC.$

$\langle 22\rangle$ $ABC(ABC)^{†} = ((B^{†}BC)^{†}(AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}$

and $(ABC)^{†}ABC = (BC)^{†}(ABB^{†})^{†}((BC)^{†}(ABB^{†})^{†})^{†}.$

$\langle 23\rangle$ $((AB)^{†}ABC(BC)^{†})^{†} = (ABC(BC)^{†})^{†}AB$ and $((AB)^{†}ABC(BC)^{†})^{†} = BC((AB)^{†}ABC)^{†}.$

$\langle 24\rangle$ $(ABC(BC)^{†})^{†} = ((AB)^{†}ABC(BC)^{†})^{†}(AB)^{†}$ and $((AB)^{†}ABC)^{†} = (BC)^{†}((AB)^{†}ABC(BC)^{†})^{†}.$

$\langle 25\rangle$ $((AB)^{†}ABC)^{†}(AB)^{†} = (BC)^{†}(ABC(BC)^{†})^{†}.$

$\langle 26\rangle$ $(ABC(BC)^{†})^{†}AB = BC((AB)^{†}ABC)^{†}.$

$\langle 27\rangle$ $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†} = (BC)^{\ast}((AB)^{†}ABC(BC)^{†})^{†}(AB)^{\ast}.$

$\langle 28\rangle$ $((BC)^{†})^{\ast}(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}(AB)^{†})^{\ast} = ((AB)^{†}ABC(BC)^{†})^{†}.$

$\langle 29\rangle$ $((AB)^{†})^{\ast}(((BC)^{†})^{\ast}(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}((AB)^{†})^{\ast})^{†}((BC)^{†})^{\ast}$ $= ((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast}.$

$\langle 30\rangle$ $((BC)^{†})^{\ast}(((AB)^{†})^{\ast}(((BC)^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast})^{†}((BC)^{†})^{\ast})^{†} ((AB)^{†})^{\ast}$

$= ((BC)^{†})^{\ast}(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}((AB)^{†})^{\ast}.$

$\langle 31\rangle$ $(AB)^{†}ABC(BC)^{†} = (AB)^{\ast}((BC)^{\ast}((AB)^{†}ABC(BC)^{†})^{†}(AB)^{\ast})^{†}(BC)^{\ast}.$

$\langle 32 \rangle$ $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}$ $= (BC)^{\ast}(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast}$

$\times ((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}(AB)^{\ast}.$

$\langle 33\rangle$ $((BC)^{†})^{\ast}(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}((AB)^{†})^{\ast}$

$= (((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((ABB^{†})^{†})^{\ast}B^{†}((BC)^{†})^{\ast}((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 34\rangle$ $(((AB)^{†})^{\ast}B^{†}BC)^{†} = ((AB)^{†}ABC)^{†}(AB)^{\ast}$ and $(ABB^{†}((BC)^{†})^{\ast})^{†} = (BC)^{\ast}(ABC(BC)^{†})^{†}.$

$\langle 35\rangle$ $((AB)^{†}ABC)^{†} = (((AB)^{†})^{\ast}B^{†}BC)^{†}((AB)^{†})^{\ast}$ and $(ABC(BC)^{†})^{†} = ((BC)^{†})^{\ast}(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 36\rangle$ $ABC(ABC)^{†} = ABB^{†}((BC)^{†})^{\ast}((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}(AB)^{†}$

and $(ABC)^{†}ABC = (BC)^{†}(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((AB)^{†})^{\ast}B^{†}BC.$

$\langle 37\rangle$ $ABC(ABC)^{†}AB = ABB^{†}((BC)^{†})^{\ast}((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}$

and $BC(ABC)^{†}ABC = (((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((AB)^{†})^{\ast}B^{†}BC.$

$\langle 38\rangle$ $ABC(ABC)^{†} = ((B^{†}((BC)^{†})^{\ast})^{†}(AB)^{†})^{†}(B^{†}((BC)^{†})^{\ast})^{†}(AB)^{†}$

and $(ABC)^{†}ABC = (BC)^{†}(((AB)^{†})^{\ast}B^{†})^{†}((BC)^{†}(((AB)^{†})^{\ast}B^{†})^{†})^{†}.$

$\langle 39\rangle$ $(B^{†}BC)^{†}(AB)^{†}ABC(((AB)^{†})^{\ast}B^{†}BC)^{†} = (B^{†}BC)^{†}(AB)^{\ast}$

and $(ABB^{†}((BC)^{†})^{\ast})^{†}ABC(BC)^{†}(ABB^{†})^{†} = (BC)^{\ast}(ABB^{†})^{†}.$

$\langle 40\rangle$ $((B^{†}BC)^{†}(AB)^{\ast})^{†}(B^{†}BC)^{†}(AB)^{†}ABC = ((AB)^{†})^{\ast}B^{†}BC$

and $ABC(BC)^{†}(ABB^{†})^{†}((BC)^{\ast}(ABB^{†})^{†})^{†} = ABB^{†}((BC)^{†})^{\ast}.$

$\langle 41\rangle$ $((AB)^{†}ABC(BC)^{†})^{†} = (((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((AB)^{†})^{\ast}$

and $((AB)^{†}ABC(BC)^{†})^{†} = ((BC)^{†})^{\ast}((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 42\rangle$ $(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†} = ((AB)^{†}ABC(BC)^{†})^{†}(AB)^{\ast}$

and $((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†} = (BC)^{\ast}((AB)^{†}ABC(BC)^{†})^{†}.$

$\langle 43\rangle$ $((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}(AB)^{\ast} = (BC)^{\ast}(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}.$

$\langle 44\rangle$ $(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†}((AB)^{†})^{\ast} = ((BC)^{†})^{\ast}((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 45\rangle$ $(BC)^{\ast}(((AB)^{†})^{\ast}B^{†}BC(BC)^{†})^{†} = ((AB)^{†}ABB^{†}((BC)^{†})^{\ast})^{†}(AB)^{\ast}.$

$\langle 46\rangle$ $(ABC)^{†} = (BC)^{\ast}((AB)^{\ast}ABC(BC)^{\ast})^{†}(AB)^{\ast}.$

$\langle 47\rangle$ $((AB)^{\ast}ABC(BC)^{\ast})^{†} = ((BC)^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast}.$

$\langle 48\rangle$ $(ABC)^{†} = (BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}.$

$\langle 49\rangle$ $((BC)^{†})^{\ast}(ABC)^{†}((AB)^{†})^{\ast} = (ABC(BC)^{\ast})^{†}ABC((AB)^{\ast}ABC)^{†}.$

$\langle 50\rangle$ $(ABC)^{†} = ((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ and $(ABC)^{†} = (BC)^{\ast}(ABC(BC)^{\ast})^{†}.$

$\langle 51\rangle$ $((AB)^{\ast}ABC)^{†} = (ABC)^{†}((AB)^{†})^{\ast}$ and $(ABC(BC)^{\ast})^{†} = ((BC)^{†})^{\ast}(ABC)^{†}.$

$\langle 52\rangle$ $(((AB)^{†})^{\ast}B^{†}BC)^{†} = (((AB)^{\ast}(AB))^{†}B^{†}BC)^{†}(AB)^{†}$

and $(ABB^{†}((BC)^{†})^{\ast})^{†} = (BC)^{†}(ABB^{†}((BC)(BC)^{\ast})^{†})^{†}.$

$\langle 53\rangle$ $(((AB)^{\ast}(AB))^{†}B^{†}BC)^{†} = (((AB)^{†})^{\ast}B^{†}BC)^{†}AB$

and $(ABB^{†}((BC)(BC)^{\ast})^{†})^{†} = BC(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 54\rangle$ $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†} = (((AB)^{\ast}AB)^{†}B^{†}((BC)^{†})^{\ast})^{†}(AB)^{†}$

and $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†} = (BC)^{†}(((AB)^{†})^{\ast}B^{†}(BC(BC)^{\ast})^{†})^{†}.$

$\langle 55\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}((BC)^{†})^{\ast})^{†} = (((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}AB$

and $(((AB)^{†})^{\ast}B^{†}(BC(BC)^{\ast})^{†})^{†} = BC(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 56\rangle$ $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†} = (BC)^{†}(((AB)^{\ast}AB)^{†}B^{†} (BC(BC)^{\ast})^{†})^{†}(AB)^{†}.$

$\langle 57\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}((BC)(BC)^{\ast})^{†})^{†} = BC(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†}AB.$

$\langle 58\rangle$ $((AB)^{\ast}ABC(BC)^{\ast})^{†} = (ABC(BC)^{\ast})^{†}((AB)^{†})^{\ast}$

and $((AB)^{\ast}ABC(BC)^{\ast})^{†} = ((BC)^{†})^{\ast}((AB)^{\ast}ABC)^{†}.$

$\langle 59\rangle$ $((AB)^{\ast}ABC)^{†} = (BC)^{\ast}((AB)^{\ast}ABC(BC)^{\ast})^{†}$ and $(ABC(BC)^{\ast})^{†} = ((AB)^{\ast}ABC(BC)^{\ast})^{†}(AB)^{\ast}.$

$\langle 60\rangle$ $((AB)^{\ast}ABC)^{†}(AB)^{\ast} = (BC)^{\ast}(ABC(BC)^{\ast})^{†}.$

$\langle 61\rangle$ $(ABC(BC)^{\ast})^{†}((AB)^{†})^{\ast} = ((BC)^{†})^{\ast}((AB)^{\ast}ABC)^{†}.$

$\langle 62\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC)^{†} = (BC)^{\ast}(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†}$

and $(ABB^{†}(BC(BC)^{\ast})^{†})^{†} = ((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 63\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†} = ((BC)^{†})^{\ast}(((AB)^{\ast}AB)^{†}B^{†}BC)^{†}$

and $((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†} = (ABB^{†}(BC(BC)^{\ast})^{†})^{†}((AB)^{†})^{\ast}.$

$\langle 64\rangle$ $(((AB)^{†})^{\ast}B^{†}(BC(BC)^{\ast})^{†})^{†}AB = BC(((AB)^{\ast}AB)^{†}B^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 65\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}((BC)^{†})^{\ast})^{†}(AB)^{†} = (BC)^{†}(((AB)^{†})^{\ast}B^{†}(BC(BC)^{\ast})^{†})^{†}.$

$\langle 66\rangle$ $((AB)^{\ast}ABC)^{†} = (AB(AB)^{\ast}ABC)^{†}AB$ and $(ABC(BC)^{\ast})^{†} = BC(ABC(BC)^{\ast}BC)^{†}.$

$\langle 67\rangle$ $(AB(AB)^{\ast}ABC)^{†} = ((AB)^{\ast}ABC)^{†}(AB)^{†}$ and $(ABC(BC)^{\ast}BC)^{†} = (BC)^{†}(ABC(BC)^{\ast})^{†}.$

$\langle 68\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC)^{†} = (((AB)^{\ast}AB(AB)^{\ast})^{†}$$B^{†}BC)^{†}$$((AB)^{†})^{\ast}$

and $(ABB^{†}(BC(BC)^{\ast})^{†})^{†} = ((BC)^{†})^{\ast}(ABB^{†}$$((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}.$

$\langle 69\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC)^{†}(AB)^{\ast} = (((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}BC)^{†}$

and $(BC)^{\ast}(ABB^{†}(BC(BC)^{\ast})^{†})^{†} = (ABB^{†}((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}.$

$\langle 70\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}((BC)^{†})^{\ast})^{†} = (((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†} ((BC)^{†})^{\ast})^{†}((AB)^{†})^{\ast}$

and $(((AB)^{†})^{\ast}B^{†}$$(BC(BC)^{\ast})^{†})^{†}$ $= ((BC)^{†})^{\ast}(((AB)^{†})^{\ast}$$B^{†}((BC)^{\ast}$$BC(BC)^{\ast})^{†})^{†}.$

$\langle 71\rangle$ $(((AB)^{\ast}AB(AB)^{\ast})^{†}$$B^{†}((BC)^{†})^{\ast})^{†} =$$(((AB)^{\ast}AB)^{†}$$B^{†}((BC)^{†})^{\ast})^{†}(AB)^{\ast}$

and $(((AB)^{†})^{\ast}B^{†}((BC)^{\ast}$$BC(BC)^{\ast})^{†})^{†} = (BC)^{\ast}$$(((AB)^{†})^{\ast}B^{†}$$(BC(BC)^{\ast})^{†})^{†}.$

$\langle 72\rangle$ $((AB)^{\ast}ABC(BC)^{\ast})^{†}$$= BC((AB)^{\ast}ABC(BC)^{\ast}BC)^{†}$

and $((AB)^{\ast}ABC(BC)^{\ast})^{†} =$$(AB(AB)^{\ast}ABC(BC)^{\ast})^{†}AB.$

$\langle 73\rangle$ $((AB)^{\ast}ABC(BC)^{\ast}BC)^{†} =$$(BC)^{†}((AB)^{\ast}ABC(BC)^{\ast})^{†}$

and $(AB(AB)^{\ast}ABC(BC)^{\ast})^{†} = ((AB)^{\ast}$$ABC(BC)^{\ast})^{†}(AB)^{†}.$

$\langle 74\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†} = BC(((AB)^{\ast}AB)^{†}B^{†}$$BC(BC)^{\ast}BC)^{†}$

and $((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†} = (AB(AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†}AB.$

$\langle 75\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast}BC)^{†} = (BC)^{†}$$(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†}$

and $((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†}(AB)^{†} = (AB(AB)^{\ast}ABB^{†}$$((BC)(BC)^{\ast})^{†})^{†}.$

$\langle 76\rangle$ $((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†} = ((BC)^{†})^{\ast}((AB)^{\ast}ABB^{†}((BC)^{\ast}$$BC(BC)^{\ast})^{†})^{†}$

and $(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†}$ $= (((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}$$BC(BC)^{\ast})^{†}((AB)^{†})^{\ast}.$

$\langle 77\rangle$ $((AB)^{\ast}ABB^{†}((BC)^{\ast}BC(BC)^{\ast})^{†})^{†} =$$(BC)^{\ast}((AB)^{\ast}ABB^{†}$$(BC(BC)^{\ast})^{†})^{†}$

and $(AB(AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†} = ((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 78\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}(BC(BC)^{\ast})^{†})^{†} = ((BC)^{†})^{\ast}(((AB)^{\ast}AB)^{†}B^{†} ((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}$

and $(((AB)^{\ast}AB)^{†}B^{†}(BC(BC)^{\ast})^{†})^{†} = (((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}(BC(BC)^{\ast})^{†})^{†}((AB)^{†})^{\ast}.$

$\langle 79\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}((BC)^{\ast}$$BC(BC)^{\ast})^{†})^{†} =$$(BC)^{\ast}(((AB)^{\ast}AB)^{†}$$B^{†}(BC(BC)^{\ast})^{†})^{†}$

and $(((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}$$(BC(BC)^{\ast})^{†})^{†} = (((AB)^{\ast}AB)^{†}B^{†}$$(BC(BC)^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 80\rangle$ $BC((AB)^{\ast}ABC(BC)^{\ast}BC)^{†} = (AB(AB)^{\ast}$$ABC(BC)^{\ast})^{†}AB.$

$\langle 81\rangle$ $((AB)^{\ast}ABC(BC)^{\ast}BC)^{†}$$(AB)^{†} = (BC)^{†}$$(AB(AB)^{\ast}ABC(BC)^{\ast})^{†}.$

$\langle 82\rangle$ $BC(((AB)^{\ast}AB)^{†}B^{†}$$BC(BC)^{\ast}BC)^{†} =$$(((AB)^{\ast}AB(AB)^{\ast})^{†}$$B^{†}BC(BC)^{\ast})^{†}$$((AB)^{†})^{\ast}.$

$\langle 83\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}$$BC(BC)^{\ast}BC)^{†}(AB)^{\ast} =$$(BC)^{†}(((AB)^{\ast}$$AB(AB)^{\ast})^{†}$$B^{†}BC(BC)^{\ast})^{†}.$

$\langle 84\rangle$ $((BC)^{†})^{\ast}$$((AB)^{\ast}ABB^{†}((BC)^{\ast}BC(BC)^{\ast})^{†})^{†} =$$(AB(AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†}AB.$

$\langle 85\rangle$ $((AB)^{\ast}ABB^{†}$$((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}$$(AB)^{†} =$$(BC)^{\ast}$$(AB(AB)^{\ast}ABB^{†}$$(BC(BC)^{\ast})^{†})^{†}.$

$\langle 86\rangle$ $(((BC)(BC)^{\ast}BC)^{†}$$(B^{†})^{\ast}((AB)^{\ast}AB)^{†})^{†}$$(BC)^{†} =$$(AB)^{†}((BC(BC)^{\ast})^{†}(B^{†})^{\ast}$$(AB(AB)^{\ast}AB)^{†})^{†}.$

$\langle 87\rangle$ $AB((BC(BC)^{\ast}BC)^{†}$$(B^{†})^{\ast}$$((AB)^{\ast}AB)^{†})^{†} =$$((BC(BC)^{\ast})^{†}(B^{†})^{\ast}$$(AB(AB)^{\ast}AB)^{†})^{†}BC.$

$\langle 88\rangle$ $((AB)^{\ast}ABC(BC)^{\ast})^{†}$$= BC(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†}AB.$

$\langle 89\rangle$ $(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†} =$$(BC)^{†}((AB)^{\ast}$$ABC(BC)^{\ast})^{†}(AB)^{†}.$

$\langle 90\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}$$BC(BC)^{\ast})^{†} =$$BC(((AB)^{\ast}$$AB(AB)^{\ast})^{†}B^{†}$$BC(BC)^{\ast}BC)^{†}$$((AB)^{†})^{\ast}.$

$\langle 91\rangle$ $(((AB)^{\ast}AB(AB)^{\ast})^{†}$$B^{†}BC(BC)^{\ast}BC)^{†} = (BC)^{†}$$(((AB)^{\ast}AB)^{†}B^{†}BC(BC)^{\ast})^{†}$$(AB)^{\ast}.$

$\langle 92\rangle$ $((AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{†} = ((BC)^{†})^{\ast}$$(AB(AB)^{\ast}ABB^{†}$$((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}AB.$

$\langle 93\rangle$ $(AB(AB)^{\ast}$$ABB^{†}((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}$ $= (BC)^{\ast}((AB)^{\ast}$$ABB^{†}(BC(BC)^{\ast})^{†})^{†}$$((AB)^{†})^{\ast}.$

$\langle 94\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}(BC(BC)^{\ast})^{†})^{†}$ = $((BC)^{†})^{\ast}((AB)^{\ast}$$AB(AB)^{\ast})^{†}B^{†}$$((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}$$((AB)^{†})^{\ast}.$

$\langle 95\rangle$ $(((AB)^{\ast}AB(AB)^{\ast})^{†}B^{†}((BC)^{\ast}BC(BC)^{\ast})^{†})^{†}$ $= (BC)^{\ast}(((AB)^{\ast}AB)^{†}$$B^{†}((BC)(BC)^{\ast})^{†})^{†}(AB)^{\ast}.$

$\langle 96\rangle$ $(AB(AB)^{\ast}ABC(BC)^{\ast}BC)^{†} = (((AB)^{\ast}AB)^2B^{†}$$BC(BC)^{\ast}BC)^{†}(AB)^{\ast}$

and $(AB(AB)^{\ast}ABC(BC)^{\ast}BC)^{†} =$$(BC)^{\ast}(AB(AB)^{\ast}ABB^{†}(BC(BC)^{\ast})^2)^{†}.$

$\langle 97\rangle$ $(((AB)^{\ast}AB)^2B^{†}BC(BC)^{\ast}BC)^{†}$$= (AB(AB)^{\ast}ABC(BC)^{\ast}BC)^{†}$$((AB)^{†})^{\ast}$

and $(AB(AB)^{\ast}ABB^{†}$$(BC(BC)^{\ast})^2)^{†} =$$((BC)^{†})^{\ast}(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†}.$

$\langle 98\rangle$ $(((AB)^{\ast}AB)^2B^{†}$$BC(BC)^{\ast}BC)^{†}(AB)^{\ast} = (BC)^{\ast}(AB(AB)^{\ast}ABB^{†}$$(BC(BC)^{\ast})^2)^{†}.$

$\langle 99\rangle$ $(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†}((AB)^{†})^{\ast} = ((BC)^{†})^{\ast}(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†}.$

$\langle 100\rangle$ $(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†} = (BC)^{\ast}(((AB)^{\ast}AB)^2B^{†}$$(BC(BC)^{\ast})^2)^{†}(AB)^{\ast}.$

$\langle 101\rangle$ $(((AB)^{\ast}AB)^2B^{†}$$(BC(BC)^{\ast})^2)^{†} = ((BC)^{†})^{\ast}(AB(AB)^{\ast}$$ABC(BC)^{\ast}BC)^{†}$$((AB)^{†})^{\ast}.$

$\langle 102\rangle$ $(BC(BC)^{†}(B^{†})^{\ast}$$(AB)^{†}AB)^{†} = ((BC)^{†}$$(B^{†})^{\ast}(AB)^{†}AB)^{†}$$(BC)^{†}$

and $(BC(BC)^{†}(B^{†})^{\ast}(AB)^{†}AB)^{†} =$$(AB)^{†}(BC(BC)^{†}(B^{†})^{\ast}$$(AB)^{†})^{†}.$

$\langle 103\rangle$ $(BC(BC)^{†}$$(B^{†})^{\ast}(AB)^{†}AB)^{†}BC = ((BC)^{†}$$(B^{†})^{\ast}(AB)^{†}AB)^{†}$

and $AB(BC(BC)^{†}(B^{†})^{\ast}$$(AB)^{†}AB)^{†} = (BC(BC)^{†}$$(B^{†})^{\ast}(AB)^{†})^{†}.$

$\langle 104\rangle$ $((AB)^{\ast}ABC(BC)^{†})^{†} = BC((AB)^{\ast}ABC)^{†}$ and $((AB)^{†}ABC(BC)^{\ast})^{†} = (ABC(BC)^{\ast})^{†}AB.$

$\langle 105\rangle$ $((AB)^{\ast}ABC)^{†} = (BC)^{†}((AB)^{\ast}ABC(BC)^{†})^{†}$ and $(ABC(BC)^{\ast})^{†} = ((AB)^{†}ABC(BC)^{\ast})^{†}(AB)^{†}.$

$\langle 106\rangle$ $((BC)^{†}(B^{†})^{\ast}((AB)^{\ast}AB)^{†})^{†} =$$(((BC)(BC)^{\ast})^{†}(B^{†})^{\ast}$$((AB)^{\ast}AB)^{†})^{†}((BC)^{†})^{\ast}$

and $((BC(BC)^{\ast})^{†}(B^{†})^{\ast}$$(AB)^{†})^{†} = ((AB)^{†})^{\ast}$$((BC(BC)^{\ast})^{†}$$(B^{†})^{\ast}((AB)^{\ast}AB)^{†})^{†}.$

$\langle 107\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}$$((BC)^{†})^{\ast})^{†} = (BC)^{†}$$(((AB)^{\ast}AB)^{†}B^{†}$$((BC)(BC)^{\ast})^{†})^{†}$

and $(((AB)^{†})^{\ast}B^{†}$$(BC(BC)^{\ast})^{†})^{†} = (((AB)^{\ast}AB)^{†}B^{†}$$(BC(BC)^{\ast})^{†})^{†}$$(AB)^{†}.$

$\langle 108\rangle$ $(((AB)^{\ast}AB)^{†}B^{†}$$(BC(BC)^{\ast})^{†})^{†} = BC(((AB)^{\ast}AB)^{†}B^{†}$$((BC)^{†})^{\ast})^{†}$

and $(((AB)^{\ast}AB)^{†}B^{†}$$(BC(BC)^{\ast})^{†})^{†} = (((AB)^{†})^{\ast}B^{†}$$(BC(BC)^{\ast})^{†})^{†}AB.$

$\langle 109\rangle$ $(ABC(B^{†}BC)^{†}(AB)^{†})^{†} = ((B^{†}BC)^{†}(AB)^{†})^{†}(ABC)^{†}$

and $(BC)^{†}(ABB^{†})^{†}ABC)^{†}$$= (ABC)^{†}((BC)^{†}$$(ABB^{†})^{†})^{†}.$

$\langle 110\rangle$ $ABC((AB)^{†}ABC)^{†}(AB)^{†}$ and $(BC)^{†}(ABC(BC)^{†})^{†}ABC$ are orthogonal projectors$.$

$\langle 111\rangle$ $ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ and $(BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC$ are orthogonal projectors$.$

$\langle 112\rangle$ $(AB)^{†}(ABC)(ABC)^{†}AB$ and $BC(ABC)^{†}(ABC)(BC)^{†}$ orthogonal projectors$.$

$\langle 113\rangle$ $ABC(ABC)^{†}$ and $AB(AB)^{\ast}$ commute$,$ and $(ABC)^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 114\rangle$ $ABC((AB)^{†}ABC)^{†}(AB)^{†}$ and $AB(AB)^{\ast}$ commute$,$ and $(BC)^{†}(ABC(BC)^{†})^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 115\rangle$ $ABC((AB)^{\ast}ABC)^{†}(AB)^{\ast}$ and $AB(AB)^{\ast}$ commute$,$ and $(BC)^{\ast}(ABC(BC)^{\ast})^{†}ABC$ and $(BC)^{\ast}BC$ commute$.$

$\langle 116\rangle$ $(AB)^{†}ABC(ABC)^{†}AB$ and $(AB)^{\ast}AB$ commute$,$ and $BC(ABC)^{†}ABC(BC)^{†}$ and $BC(BC)^{\ast}$ commute$.$

$\langle 117\rangle$ $AB((B^{†}BC)^{†}$$(AB)^{†}AB)^{†}(B^{†}BC)^{†}$$(AB)^{†}$ and $(BC)^{†}(ABB^{†})^{†}(BC(BC)^{†}$$(ABB^{†})^{†})^{†}BC$

are orthogonal projectors$.$

$\langle 118\rangle$ $((AB)^{†})^{\ast}((B^{†}BC)^{†}$$((AB)^{\ast}AB)^{†})^{†}$$(B^{†}BC)^{†}(AB)^{†}$

and $(BC)^{†}(ABB^{†})^{†}$$((BC(BC)^{\ast})^{†}(ABB^{†})^{†})^{†}$$((BC)^{†})^{\ast}$ are orthogonal projectors$.$

$\langle 119\rangle$ $(AB)^{†}((B^{†}BC)^{†}$$(AB)^{†})^{†}$$(B^{†}BC)^{†}$$(AB)^{†}AB$ and $BC(BC)^{†}(ABB^{†})^{†}$$((BC)^{†}(ABB^{†})^{†})^{†}$$(BC)^{†}$ are orthogonal projectors$.$

$\langle 120\rangle$ $((B^{†}BC)^{†}(AB)^{†})^{†}(B^{†}BC)^{†}(AB)^{†}$ and $(AB(AB)^{\ast})^{†}$ commute$,$ and $(BC)^{†}(ABB^{†})^{†}((BC)^{†}(ABB^{†})^{†})^{†}$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 121\rangle$ $AB((B^{†}BC)^{†}(AB)^{†}AB)^{†}(B^{†}BC)^{†}(AB)^{†}$ and $(AB(AB)^{\ast})^{†}$ commute$,$

and $(BC)^{†}(ABB^{†})^{†}(BC(BC)^{†}(ABB^{†})^{†})^{†}BC$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 122\rangle$ $((AB)^{†})^{\ast}((B^{†}BC)^{†}$$((AB)^{\ast}AB)^{†})^{†}$$(B^{†}BC)^{†}$$(AB)^{†}$ and $(AB(AB)^{\ast})^{†}$ commute$,$

and $(BC)^{†}(ABB^{†})^{†}((BC(BC)^{\ast})^{†}(ABB^{†})^{†})^{†}((BC)^{†})^{\ast}$ and $((BC)^{\ast}BC)^{†}$ commute$.$

$\langle 123\rangle$ $(AB)^{†}((B^{†}BC)^{†}(AB)^{†})^{†}((B^{†}BC)^{†}(AB)^{†})AB$ and $((AB)^{\ast}AB)^{†}$ commute$,$

and $BC((BC)^{†}(ABB^{†})^{†})((BC)^{†}(ABB^{†})^{†})^{†}(BC)^{†}$ and $(BC(BC)^{\ast})^{†}$ commute$.$

$\langle 124\rangle$ $ABC(B^{†}BC)^{†}(AB)^{†}$ and $(BC)^{†}(ABB^{†})^{†}ABC$ are EP$.$

$\langle 125\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\}$$\ni ((AB)^{†}ABC)^{†}(AB)^{\ast}$

and $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni (BC)^{\ast}(ABC(BC)^{†})^{†}.$

$\langle 126\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\}$$\ni (((AB)^{\ast}AB)^{†}B^{†}BC)^{†}$$(AB)^{†}$

and $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\}$$\ni (BC)^{†}(ABB^{†}$$(BC(BC)^{\ast})^{†})^{†}.$

$\langle 127\rangle$ $\{((AB)^{†}ABC)^{(1, 2, 3)}\} \ni (((AB)^{†})^{\ast}$$B^{†}BC)^{†}((AB)^{†})^{\ast}$

and $\{(ABC(BC)^{†})^{(1, 2, 4)}\} \ni ((BC)^{†})^{\ast}(ABB^{†}$$((BC)^{†})^{\ast})^{†}.$

$\langle 128\rangle$ $\{(((AB)^{\ast}AB)^{†}B^{†}BC)^{(1, 2, 3)}\} \ni (((AB)^{†})^{\ast}B^{†}BC)^{†}AB$

and $\{(ABB^{†}(BC(BC)^{\ast})^{†})^{(1, 2, 4)}\} \ni BC(ABB^{†}((BC)^{†})^{\ast})^{†}.$

$\langle 129\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\} \ni (B^{†}BC)^{†}$$((AB)^{†}ABC(B^{†}BC)^{†})^{†}(AB)^{\ast}$

and $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\} \ni$$(BC)^{\ast}((ABB^{†})^{†}$$ABC(BC)^{†})^{†}$$(ABB^{†})^{†}.$

$\langle 130\rangle$ $\{(((AB)^{†})^{\ast}B^{†}BC)^{(1, 2, 3)}\}$$\ni (B^{†}BC)^{\ast}$$(((AB)^{\ast}AB)^{†}B^{†}$$BC(B^{†}BC)^{\ast})^{†}(AB)^{†}$

and $\{(ABB^{†}((BC)^{†})^{\ast})^{(1, 2, 4)}\}$$\ni (BC)^{†}((ABB^{†})^{\ast}ABB^{†}$$(BC(BC)^{\ast})^{†})^{†}$$(ABB^{†})^{\ast}.$

$\langle 131\rangle$ $\{((AB)^{†}$$ABC(B^{†}BC)^{†})^{(1, 2, 3)}\} \ni B^{†}$$BC(((AB)^{†})^{\ast}B^{†}BC)^{†}((AB)^{†})^{\ast}$

and $\{((ABB^{†})^{†}$$ABC(BC)^{†})^{(1, 2, 4)}\} \ni((BC)^{†})^{\ast}(ABB^{†}$$((BC)^{†})^{\ast})^{†}ABB^{†}.$

$\langle 132\rangle$ $\{(((AB)^{\ast}AB)^{†}$$B^{†}BC(B^{†}BC)^{\ast})^{(1, 2, 3)}\}$$\ni ((B^{†}BC)^{†})^{\ast}$$(((AB)^{†})^{\ast}B^{†}BC)^{†}AB$

and $\{((ABB^{†})^{\ast}ABB^{†}(BC(BC)^{\ast})^{†})^{(1, 2, 4)}\} \ni$$BC(ABB^{†}((BC)^{†})^{\ast})^{†}$$((ABB^{†})^{†})^{\ast}.$

$\langle 133\rangle$ $(AB(AB)^{\ast}ABC)(AB(AB)^{\ast}ABC)^{†}$$= (ABC)(ABC)^{†}$

and $(ABC(BC)^{\ast}BC)^{†}$$(ABC(BC)^{\ast}BC) =$$(ABC)^{†}(ABC).$

$\langle 134\rangle$ ${\mathscr R}((ABC)^{†}) =$${\mathscr R}((BC)^{†}((AB)^{†}$$ABC(BC)^{†})^{†}(AB)^{†})$

and ${\mathscr R}(((ABC)^{†})^{\ast})$$= {\mathscr R}(((BC)^{†}((AB)^{†}$$ABC(BC)^{†})^{†}$$(AB)^{†})^{\ast}).$

$\langle 135\rangle$ ${\mathscr R}(AB(AB)^{\ast}ABC) =$${\mathscr R}(ABC)$ and ${\mathscr R}((BC)^{\ast}BC(ABC)^{\ast})$$= {\mathscr R}((ABC)^{\ast}).$

$\langle 136\rangle$ $r[AB(AB)^{\ast}ABC, \, ABC]$$= r[(BC)^{\ast}BC(ABC)^{\ast},$$\, (ABC)^{\ast}] = r(ABC).$

$\langle 137\rangle$ $r[((AB)^{\ast}AB(AB)^{\ast})^{†}$$B^{†}BC, \, ((AB)^{†})^{\ast}B^{†}BC]$$= r(((AB)^{†})^{\ast}B^{†}BC)$

and $r[(BC(BC)^{\ast}BC)^{†}(ABB^{†})^{\ast},$$\, (BC)^{†}(ABB^{†})^{\ast}] =$$r((BC)^{†}(ABB^{†})^{\ast}).$

Theorem 4.5. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following three statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = (BC)^{†}$$((AB)^{†}ABC(BC)^{†})^{†}$$(AB)^{†}.$

$\langle 2\rangle$ $(ABC)^{†} = (BC)^{†}((AB)^{†}ABB^{†} + B^{†}BC$$(BC)^{†} -$$B^{†})^{†}(AB)^{†}.$

$\langle 3\rangle$ $(((AB)^{†})^{\ast}B^{†}((BC)^{†})^{\ast})^{†} = (BC)^{\ast}((AB)^{†}ABB^{†}$$+ B^{†}BC(BC)^{†} -$$B^{†})^{†}(AB)^{\ast}.$

Corollary 4.6. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{n \times p},$ and $C \in {\mathbb C}^{p \times q}.$ Then the following two statements are equivalent$:$

$\langle 1\rangle$ $(ABC)^{†} = (BC)^{†}((AB)^{†}ABC(BC)^{†})^{†}(AB)^{†}.$

$\langle 2\rangle$ $(((AB)^{\ast}AB)^{1/2}B^{†}(BC(BC)^{\ast})^{1/2})^{†}$ $= ((BC(BC)^{\ast})^{1/2})^{†}$$((AB)^{†}ABC(BC)^{†})^{†}$$(((AB)^{\ast}AB)^{1/2})^{†}.$

If $(AB)^{†}ABC(BC)^{†} = B^{†}$, then (1.18) is reduced to $(ABC)^{†} = (BC)^{†}B(AB)^{†}$. In this case, it is easy to obtain a group of equivalent statements associated with the nested ROL from Theorem 4.4, which we leave for the reader.

As applications of the results in Sections 3 and 4, we are able to establish and simplify many other types of matrix expressions and matrix equalities that involve generalized inverses. Here, we mention a convenient way to rewrite the sum $A + B$, where $A, \, B \in {\mathbb C}^{m \times n}$, as the following products of triple block matrices:

In these situations, applying the preceding theorems and corollaries to the triple matrix products on the right-hand sides of the four matrix equalities will correspondingly yield several groups of results on the generalized inverses of the sum of two matrices. But we omit them here due to space limits.

5.   Conclusions

We have collected and established a myriad of known and novel nested ROLs for generalized inverses of triple matrix matrices. These ROLs and their variations can be classified as concrete examples in the establishment and characterization of various matrix equalities of generalized inverses, so that they provide highly informative accounts of a variety of current researches concerning equalities for matrices and their generalized inverses, and of course can be used as analytic tools to deal adequately with various theoretical and computational problems in the theory of generalized inverses of matrices.

As demonstrated in the preceding sections, ROLs for generalized inverses of multiple products of singular matrices can reasonably be constructed in numerous regular and nested forms, which seem quite complicated in contrast with the ordinary inverses of nonsingular matrices. To illustrate, we present several examples of nested ROLs for the Moore–Penrose generalized inverses of products of four matrices as follows

Recall that classification becomes a common theme across all areas of mathematics as various problems, formulas, results, and facts in each field increase gradually. Thus, the equivalence classifications of the ROLs have naturally been proposed and have become one of the challenging but fruitful working areas in the theory of generalized inverses. Nevertheless, the past several decades have seen magnificent breakthroughs via successful adaptation of the matrix rank method and the block matrix method in the investigation of ROLs. By now, the classification program has proven a resounding success in the establishment of matrix equalities composed of generalized inverses, in particular, it has been realized that all ROLs for generalized inverses of multiple matrix products can be divided into certain groups, for which we can approach jointly and obtain many equivalent facts with great efficiency.

Acknowledgments

The author wishes to thank the editor and anonymous referees for their helpful comments and suggestions on an earlier version of this article.

Conflict of interest

The author declares no conflict of interest.