Characterization of frame vectors for $\mathcal{A}$-group-like unitary systems on Hilbert $C^{\ast}$-modules

1.   Introduction

Frames in Hilbert spaces were first introduced by Duffin and Schaeffer ^[7] to study the nonharmonic Fourier series. The frame theory has experienced tremendous development in the past decades. It was motivated by engineering applications and pure mathematics ^[1,3,4,5]. Frank and Larson extended the concept of frame in Hilbert spaces to countably generated $C^{\ast}$-modules and investigated its main features ^[8,9]. While a Hilbert $C^{\ast}$-module is a broader concept than a Hilbert space, there are several distinctions between the two. There are instances where adjoint operators are not present for certain bounded operators on Hilbert $C^{\ast}$-modules. It should be noted that although some definitions and results of modular frames may look similar to their Hilbert space frame counterparts, the complexity of the Hilbert $C^{\ast}$-module structure makes it difficult to simplify Hilbert space frames into generalizations. We refer to more discussion on essential differences between frames in Hilbert spaces and in $C^{\ast}$-modules ^[8,9,12]. Several generalizations of frames have been defined in Hilbert $C^{\ast}$-modules, as documented in ^{[13,15,18,21]}.

In recent years, there has been growing evidence that modular frames are closely related to other research areas, such as wavelet frame construction ^[16,20]. Considering the theory and applications of structured frames in Hilbert spaces, such as Gabor and wavelet frames, have always been the main focus of Hilbert space frame theory, we believe the structured modular frames may be well suited for specific applications in terms of theoretical or application properties. Dai and Larson ^[6] and Han and Larson ^[11] have introduced a fresh perspective on analyzing structured frames by examining frame vectors for unitary systems. This approach has garnered significant interest in frame theory. Building on the concept of group-like unitary systems in Hilbert spaces proposed in ^[10], we have introduced the idea of $\mathcal{A}$-group-like unitary systems in Hilbert $C^{\ast}$-modules. The purpose of this paper is to explore the frames that are created by $\mathcal{A}$-group-like unitary systems.

The organizational structure of this paper is as follows. In Section 2, we state some notations and preliminaries, and introduce the notion of an $\mathcal{A}$-group-like unitary system $\mathcal{U}$ on Hilbert $C^{\ast}$-modules. Section 3 describes the dilation theory of frames induced by $\mathcal{U}$ on Hilbert $C^{\ast}$-modules. In Section 4, characterization of the complete Bessel vectors for $\mathcal{U}$ is obtained in terms of certain class of operators in $\mathcal{U}''$. Finally, we prove that the frame multiplicity of $\mathcal{U}$ is always finite in Section 5.

2.   Preliminaries

In the following, we will review the fundamental definitions and characteristics of Hilbert $C^{\ast}$-modules and their frames. Suppose that $\mathcal{A}$ is a unital $C^{\ast}$-algebra and $\mathcal{H}$ is a left $\mathcal{A}$-module. $\mathcal{H}$ is a pre-Hilbert $\mathcal{A}$-module if $\mathcal{H}$ is equipped with an $\mathcal{A}$-valued inner product $\langle\cdot, \cdot \rangle:\mathcal{H}\times \mathcal{H}\rightarrow \mathcal{A}$ such that the following properties hold:

(1) $\langle x, x\rangle\geq 0$ for all $x\in \mathcal{H}$ and $\langle x, x\rangle = 0$ if and only if $x = 0$.

(2) $\langle ax+y, z\rangle = a\langle x, z\rangle+\langle y, z\rangle$ for every $a\in \mathcal{A}$, every $x, y, z\in \mathcal{H}$.

(3) $\langle x, y\rangle = \langle y, x\rangle^{\ast}$ for every $x, y\in \mathcal{H}$.

For every $x\in \mathcal{H}$, we define

If $\mathcal{H}$ is complete with $\|\cdot\|$, it is called a Hilbert $\mathcal{A}$-module (or a Hilbert $C^{\ast}$-module over $\mathcal{A}$). For every $a\in \mathcal{A}$, we have $|a| = (a^{\ast}a)^{\frac{1}{2}}$, and the $\mathcal{A}$-valued norm on $\mathcal{H}$ is defined by $|x| = \langle x, x\rangle^{\frac{1}{2}}$.

Lemma 2.1 (^[17]). Assume $\mathcal{H}$, $\mathcal{K}$ are Hilbert $\mathcal{A}$-modules, and $T:\mathcal{H}\rightarrow \mathcal{K}$ is a linear map. Then the conditions listed below are equivalent:

(1) $T$ is both bounded and $\mathcal{A}$-linear.

(2) There is $d\geq 0$ such that $\langle Tz, Tz\rangle \leq d\langle z, z\rangle, z\in \mathcal{H}$.

A Hilbert $\mathcal{A}$-module $\mathcal{H}$ is called finitely generated if there is a finite set $\{z_{1}, \cdots z_{n}\}$ of $\mathcal{H}$ such that $z = \sum^{n}_{i = 1}a_{i}z_{i}, a_{i}\in \mathcal{A}$ for each $z\in \mathcal{H}$, if the set of generators is countable, it is referred to as countably generated.

Let $\mathcal{H}$ and $\mathcal{K}$ be two Hilbert $\mathcal{A}$-modules. We say that the operator $T:\mathcal{H}\rightarrow \mathcal{K}$ is adjointable, if there is another one $T^{\ast}:\mathcal{K} \rightarrow \mathcal{H}$ such that $\langle Tz, y\rangle=\langle z, T^{\ast}y\rangle, \; \forall z\in \mathcal{H}, y\in \mathcal{K}$. It's important to note that an adjointable operator is both $\mathcal{A}$-linear and bounded by default. Any families of adjointable operators from $\mathcal{H}$ to $\mathcal{K}$ are referred to as ${\rm End}_{\mathcal{A}}^{\ast}(\mathcal{H}, \mathcal{K})$. If $\mathcal{K}$ equals $\mathcal{H}$, it is shortened to ${\rm End}_{\mathcal{A}}^{\ast}(\mathcal{H})$. An operator $T\in {\rm End}_{\mathcal{A}}^{\ast}(\mathcal{H}, \mathcal{K})$ is classified as a unitary if $TT^{\ast}=I_{\mathcal{K}}$ and $T^{\ast}T=I_{\mathcal{H}}$. For further information on Hilbert $C^{\ast}$-modules, please refer to ^[14].

In this paper, $\mathcal{A}$ represents a unital $C^{\ast}$-algebra. $\mathcal{H}$ and $\mathcal{K}$ are Hilbert $\mathcal{A}$-modules that are either finitely or countably generated. Additionally, $\mathbb{I}$ is a countable index set. If $T$ is a bounded $\mathcal{A}$-linear operator, then ${\rm ran}T$, ${\rm ker}T$ and $T^{\ast}$ refer to its range space, nullspace and adjoint, respectively.

In line with reference ^[9], we will define frames in $C^{\ast}$-modules.

Definition 2.2. A sequence $\{x_{i}\}_{i\in \mathbb{I}}$ of elements in $\mathcal{H}$ is said to be a frame if there exist two constants $A, B > 0$ such that

for every $x\in \mathcal{H}$. The constants $A$ and $B$ are called frame bounds. The frame $\{x_{i}\}_{i\in \mathbb{I}}$ is considered tight if $A = B$ and Parseval if $A = B = 1$. Likewise, if $\{x_{i}\}_{i\in \mathbb{I}}$ only satisfies the upper bound condition in (2.1), then $\{x_{i}\}_{i\in \mathbb{I}}$ is called a Bessel sequence with bound $B$. A frame is considered standard if the norm converges in the middle sum of (2.1).

We will consider only the standard (Parseval) frames and the standard Bessel sequences. This paper will focus on finitely or countably generated Hilbert $C^{\ast}$-modules, as the Kasparov Stabilization Theorem establishes a frame available for every such module. For a standard Bessel sequence $\{x_{i}\}_{i\in \mathbb{I}}$, the analysis operator $T$ is an $\mathcal{A}$-linear bounded adjointable operator from $\mathcal{H}$ to $l^{2}(\mathcal{A})$ defined by

The adjoint operator $T^{\ast}: l^{2}(\mathcal{A}) \rightarrow \mathcal{H}$ is defined by

and it is called the synthesis operator of $\{x_{i}\}_{i\in \mathbb{I}}$. In fact, if $\{x_{i} \}_{i\in \mathbb{I} }$ is a standard frame for $\mathcal{H}$, the frame operator

is a well-defined, positive, adjointable and invertible operator and the following reconstruction formula

holds for all $x \in \mathcal{H}$. Moreover, $\{S^{-1} x_{i}\}_{i\in \mathbb{I}}$ is also a frame for $\mathcal{H}$ and is called the canonical dual of $\{x_{i}\}_{i\in \mathbb{I}}$.

An orthonormal system in $\mathcal{H}$, denoted by $\{f_{i}\}_{i\in \mathbb{I} }$, is a family of vectors such that $\langle f_{i}, f_{j}\rangle = 0, \forall i\neq j$ and $\langle f_{i}, f_{i}\rangle = 1, \forall i\in \mathbb{I}$. If an orthonormal system generates a dense submodule of $\mathcal{H}$, it is called an orthonormal basis for $\mathcal{H}$.

A unitary system $\mathcal{U}$ on $\mathcal{H}$ is a subset of all unitary operators in ${\rm End}^{\ast}_{\mathcal{A}}(\mathcal{H})$ that includes the identity operator $I$. For a unitary system $\mathcal{U}$, a vector $x\in \mathcal{H}$ is called a Parseval frame vector (resp. frame vector or Bessel vector) for $\mathcal{U}$ if $\mathcal{U}x$ forms a Parseval frame (resp. frame or Bessel sequence) for ${\rm \overline{span}} \{Ux: U\in \mathcal{U}\}$. When ${\rm \overline{span}} \{Ux: U\in \mathcal{U}\}=\mathcal{H}$, the frame vector is said to be complete. If $\mathcal{U}x$ is an orthonormal basis for $\mathcal{H}$, $x$ is a complete wandering vector for $\mathcal{U}$. The set of all complete wandering vectors for $\mathcal{U}$ is represented by $\mathcal{W}(\mathcal{U})$.

Suppose that $\mathcal{A}$ is a unital $C^{\ast}$-algebra with identity $1$. We use $\mathcal{Z}(\mathcal{A}) = \{a\in \mathcal{A}: ab = ba, \; \forall\; b\in \mathcal{A}\}$ to represent the centre of $\mathcal{A}$. Inspired by the concept of group-like unitary systems in Hilbert spaces, which was proposed in ^[10], we introduce the notion of $\mathcal{A}$-group-like unitary systems in Hilbert $C^{\ast}$-modules. Let ${\rm group}(\mathcal{U})$ be the group generated by $\mathcal{U}$ and $\mathbb{A} = \{a\in \mathcal{Z}(\mathcal{A}): aa^{\ast} = a^{\ast}a = 1\}$. We call $\mathcal{U}$ an $\mathcal{A}$-group-like unitary system if

and $\mathcal{U}$ is $\mathbb{A}$-linearly independent, i.e., $\mathbb{A}U \neq \mathbb{A}V$ as long as $U$ and $V$ are different elements of $\mathcal{U}$. Then there exists a function $f: {\rm group}(\mathcal{U})\rightarrow \mathbb{A}$ and a mapping $\tau: {\rm group}(\mathcal{U})\rightarrow \mathcal{U}$ such that $W = f(W)\tau(W)$ for all $W\in {\rm group}(\mathcal{U})$. It is essential to provide clear definitions of $f$ and $\tau$. When $W$ is equal to $a_{1}U_{1}$ and $a_{2}U_{2}$ with $U_{1}, U_{2}\in \mathcal{U}, a_{1}, a_{2}\in \mathbb{A}$, it can be determined that $a_{1}$ equals $a_{2}$ and $U_{1}$ equals $U_{2}$ since $\mathcal{U}$ is $\mathbb{A}$-linearly independent. When dealing with the mappings $f$ and $\tau$, there is a fundamental property that we can establish. The proof of this property is similar to the one presented in the literature's appendix ^[10].

Proposition 2.3. Let $\mathcal{U}, f$ and $\tau$ be as above. Then

(1) $\tau(U) = U, f(U) = 1, U\in \mathcal{U}$.

(2) $f(U\tau(VW))f(VW) = f(\tau(UV)W)f(UV), U, V, W\in {\rm group}(\mathcal{U})$.

(3) $\tau(U\tau(VW)) = \tau(\tau(UV)W), U, V, W\in {\rm group}(\mathcal{U})$.

(4) If $V, W\in {\rm group}(\mathcal{U})$, then

(5) Let $V, W\in \mathcal{U}$. Then the mappings from $\mathcal{U}$ to $\mathcal{U}$ listed below are all injective:

Proof. The statement (1) is trivial. Statements (2) and (3) come from the equality

and the assumption that $\mathcal{U}$ is $\mathbb{A}$-linear independent.

Since $\tau(\tau(SV^{-1})V) = \tau(S\tau(V^{-1}V)) = \tau(S) = S, \forall S\in \mathcal{U}$, which implies that the first equality in (4) holds. Notice

Hence $\tau(V\tau(S^{-1}V)^{-1}) = S$ since $S\in \mathcal{U}$. Similarly,

Thus the rest of (4) holds.

For (5), suppose that $\tau(VU_{1}^{-1}W) = \tau(VU_{2}^{-1}W)$ for some $U_{1}, U_{2}\in \mathcal{U}$. Then

where $f^{\ast}(VU_{1}^{-1}W)$ denotes the adjoint of $f(VU_{1}^{-1}W)$. Hence

So $U_{1} = U_{2}$, which implies that $U\rightarrow \tau(VU^{-1}W)$ is injective. Similarly, the rest of (5) hold. □

The commutant of a subset $S$ in ${\rm End}^{\ast}_{\mathcal{A}}(\mathcal{H})$ is $S' = \{A\in {\rm End}^{\ast}_{\mathcal{A}}(\mathcal{H}):AS = SA, S\in \mathcal{S} \}$. The local commutant of a nonzero vector $\xi\in \mathcal{H}$ is denoted as

The Hilbert $C^{\ast}$-module $l_{\mathcal{U}}^{2}(\mathcal{A})$ is defined by

The standard orthonormal basis of $l_{\mathcal{U}}^{2}(\mathcal{A})$ is represented by $\{\chi_{U}\}_{U\in \mathcal{U}}$. Here, $\chi_{U}$ is 1 at $U$ and 0 everywhere else. For each $U\in \mathcal{U}$, we define

$L$ and $R$ here are the left and right regular representations of $\mathcal{U}$.

To prove our main results, we need the following:

Lemma 2.4. Suppose that $\mathcal{A}$ is a commutative unital $C^{\ast}$-algebra. Then

where $\mathcal{L} = \{L_{U}: U\in\mathcal{U}\}''$ and $\mathcal{R} = \{R_{U}: U\in\mathcal{U}\}''.$

Proof. It is easy to check $L_{U}R_{V} = R_{V}L_{U}, \forall \; U, V\in \mathcal{U}$. Let $T\in \mathcal{L}'$ and $S\in \mathcal{R}'$. To prove this lemma, it is sufficient to prove $TS = ST$. Write

Then for any $V\in \mathcal{U}$, we have

The other side,

Since $\mathcal{A}$ is a commutative $C^{\ast}$-algebra, which implies that $ST\chi_{V} = TS\chi_{V}$, and so $ST = TS$. □

3.   Dilation of complete Parseval frame vectors for $\mathcal{A}$-group-like unitary systems

This section aims to prove the dilation property of complete Parseval frame vectors for $\mathcal{U}$. A simple lemma below will be used to prove Theorem 3.2.

Lemma 3.1. Suppose that $\mathcal{U}$ admits a complete Parseval frame vector $\eta$. Then $\mathcal{U}$ is unitarily equivalent to $\{L_{U}|_{\mathcal{K}}:U\in \mathcal{U}\}$, where $\mathcal{K} = T_{\eta}(\mathcal{H})$ and $T_{\eta}:\mathcal{H} \rightarrow l_{\mathcal{U}}^{2}(\mathcal{A})$ is the analysis operator defined by

Proof. Because $\eta$ is a complete Parseval frame vector for $\mathcal{U}$, it is easy to check that $T_{\eta}$ is an adjointable isometry. By Theorems 15.3.5 and 15.3.8 in ^[19] we have

Consider $P$ as the orthogonal projection from $l_{\mathcal{U}}^{2}(\mathcal{A})$ onto $T_{\eta}(\mathcal{H})$. Then we have $T_{\eta}U\eta = P\chi_{U}.$ In fact, let $V\in \mathcal{U}$,

We first show that $T_{\eta}T_{\eta}^{\ast} = P$. In fact, for every $z\in l_{\mathcal{U}}^{2}(\mathcal{A})$, we can assume that $z = \sum_{U\in \mathcal{U}}a_{U}\chi_{U}$ for some $a_{U}\in \mathcal{A}$. Then we have

Moreover, for each $V\in \mathcal{U}$, we have

Therefore $PL_{V} = T_{\eta}VT_{\eta}^{\ast}$. For each $V\in \mathcal{U}$, we also have

Thus $L_{U}T_{\eta} = T_{\eta}U$. Finally, for any $U\in \mathcal{U}$,

Hence $P\in \{L_{U}:U\in \mathcal{U}\}'$. Then

i.e., $PT_{\eta}U = PL_{U}PT_{\eta}$. Since $T_{\eta}$ is an isometry, we have

and $PT_{\eta}$ is surjective, then $PT_{\eta}$ is a unitary operator. Let $W = PT_{\eta}$, we have $WU = PL_{U}W$ for every $U\in \mathcal{U}$. Since $P(l_{\mathcal{U}}^{2}(\mathcal{A})) = T_{\eta}(\mathcal{H}) = \mathcal{K}$ and $P\in \{L_{U}:U\in \mathcal{U}\}'$, it is found that $\mathcal{U}$ is unitarily equivalent to $\{L_{U}|_{\mathcal{K}}:U\in \mathcal{U}\}$, where $\mathcal{K} = T_{\eta}(\mathcal{H})$. □

Next, we give the dilation theorem of frames induced by $\mathcal{A}$-group-like unitary systems on Hilbert $C^{\ast}$-modules.

Theorem 3.2. Let $\mathcal{U}_{1}$ be an $\mathcal{A}$-group-like unitary system on a finitely or countable generated Hilbert $\mathcal{A}$-module $\mathcal{H}$ over a unital $C^{\ast}$-algebra $\mathcal{A}$, and let $\xi$ be a complete Parseval frame vector for $\mathcal{U}_{1}$. Then there exists a finitely or countable generated Hilbert $\mathcal{A}$-module $\mathcal{M}$, an $\mathcal{A}$-group-like unitary system $\mathcal{U}_{2}$ on $\mathcal{M}$ and a complete Parseval frame vector $\eta$ for $\mathcal{U}_{2}$ such that $\xi\oplus\eta$ is a complete wandering vector for $\mathcal{U}_{1}\oplus \mathcal{U}_{2}$ on $\mathcal{H}\oplus \mathcal{M}$.

Proof. Let $\mathcal{K} = l_{\mathcal{U}_{1}}^{2}(\mathcal{A})$. For each $U_{1}\in \mathcal{U}_{1}$, let $L_{U_{1}}$ be the left regular representation. Consider $\mathcal{U} = \{L_{U_{1}}: U_{1}\in \mathcal{U}_{1}\}$. Now we consider the analysis operator associated with $\xi$ which is denoted by $W$. Then $W$ is an adjointable isometry and has closed range. Let $P$ be the orthogonal projection onto ${\rm ran }W$. We have $P\chi_{U_{1}} = WU_{1}\xi$. Let $U_{1} = I$. We have $P\chi_{I} = W\xi$.

We first show that $P\in \mathcal{U}'$. For every $z\in \mathcal{H}, U_{1}, V_{1}\in \mathcal{U}_{1}$,

Then $L_{ V_{1}}W = WV_{1}$ for all $V_{1}\in \mathcal{U}_{1}$. Therefore, we have

It follows that $P\in \mathcal{U}'$ and also $P^{\bot}\in \mathcal{U}'$.

Let $\mathcal{U}_{2} = \{P^{\bot}L_{U_{1}}P^{\bot}:U_{1}\in \mathcal{U}_{1}\}, \mathcal{M} = P^{\bot}\mathcal{H}, \eta = P^{\bot}\chi_{I}$. The test that $\mathcal{U}_{2}$ is an $\mathcal{A}$-group-like unitary system on $\mathcal{M}$ is easy.

Now, we show that $\eta$ is a complete Parseval frame vector for $\mathcal{U}_{2}$. In fact, for every $z\in \mathcal{H}$,

The conclusion is proved by this. Finally, we show $\xi\oplus\eta$ is a complete wandering vector for $\mathcal{U}_{1}\oplus \mathcal{U}_{2}$ on $\mathcal{H}\oplus \mathcal{M}$. Then for every $U_{1}, V_{1}\in \mathcal{U}_{1}$, we have

For $z_{1}\in \mathcal{H}, z\in \mathcal{H}\oplus \mathcal{M}$, we also have

The conclusion is proved. □

Then we have

Corollary 3.3. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ such that $\mathcal{W}(\mathcal{U})\neq \emptyset$ and let $\xi$ be a Parseval frame vector for $\mathcal{U}$. Then there exists a Parseval frame vector $\eta$ for $\mathcal{U}$ such that $\xi\oplus\eta$ is a complete wandering vector for $\mathcal{U}$ on $\mathcal{H}$.

4.   The complete Bessel vectors

Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system on $\mathcal{H}$ over a unital commutative $C^{\ast}$-algebra $\mathcal{A}$, and let $\xi\in \mathcal{H}$ be a complete Bessel vector. The analysis operator with respect to $\xi$ defined by

is adjointable and fulfils $T^{\ast}_{\xi}\chi_{U} = U\xi$ by ^[9]. It is well known $\{U\xi\}_{U\in \mathcal{U}}$ is a frame if and only if $T^{\ast}_{\xi}$ is surjective ^[2]. In what follows, let $B_{\mathcal{U}}$ be the set of all complete Bessel vectors for $\mathcal{U}$.

Proposition 4.1. Suppose that $\mathcal{U}$ is an $\mathcal{A}$-group-like unitary system on a Hilbert $\mathcal{A}$-module $\mathcal{H}$. If $\xi\in \mathcal{H}$ such that $\mathcal{U}\xi$ is a generator of $\mathcal{H}$, then $C_{\xi}(\mathcal{U}) = \mathcal{U}'$. Moreover, $T_{\eta}^{\ast}T_{\zeta}\in \mathcal{U}'$ for all $\eta, \zeta\in B_{\mathcal{U}}$.

Proof. The inclusion $"\supseteq"$ is trivial. Let $T\in C_{\xi}(\mathcal{U})$ be arbitrary. Then

Since $\mathcal{U}\xi$ is the generator of $\mathcal{H}$, which implies that $TU = UT$. Thus $C_{\xi}(\mathcal{U}) \subseteq\mathcal{U}'$.

For the second part, for every $V\in \mathcal{U}$, we have

where $z\in \mathcal{H}, \eta, \zeta\in B_{\mathcal{U}}$. So we have $T_{\eta}^{\ast}T_{\zeta}\in \mathcal{U}'$ for all $\eta, \zeta\in B_{\mathcal{U}}$. □

The following proposition can be the analogue of the corresponding result for frames in Hilbert space ^[6].

Proposition 4.2. Suppose that $\mathcal{H}$ has orthonormal bases and $\xi\in \mathcal{W}(\mathcal{U})$. Then $\eta\in B_{\mathcal{U}}$ if and only if there is an adjointable operator $T\in C_{\xi}(\mathcal{U})$ such that $\eta = T\xi$.

Proof. Suppose that $\eta = T\xi$ for some adjointable operator $T\in C_{\xi}(\mathcal{U})$. Then for any $z\in \mathcal{H}$,

where the last inequality we use Lemma 2.1. Then $\eta\in B_{\mathcal{U}}$.

Now, suppose that $\eta\in B_{\mathcal{U}}$. The analysis operators with respect to $\xi$ and $\eta$ are represented by $T_{\xi}$ and $T_{\eta}$, respectively. Let $T = T^{\ast}_{\eta}T_{\xi}$. Then $T$ is adjointable. We now show that $\eta = T\xi$ and $T\in C_{\xi}(\mathcal{U})$. By Proposition 4.1, $T\in \mathcal{U}'\subseteq C_{\xi}(\mathcal{U})$. For any $U\in \mathcal{U}$,

Let $U = I$. Then $T\xi = \eta$. □

To parametrize the collection of all complete Bessel vectors for $\mathcal{U}$, we utilize Lemma 2.4 which introduces a natural $\mathcal{A}$-conjugate linear isomorphism $\pi$ from $\mathcal{L}$ onto $\mathcal{L}'$. This is accomplished by defining

In particular, $\pi(A)\chi_{I} = A^{\ast}\chi_{I}, A\in \mathcal{L}$. The following is a parametrization of the set of all complete Bessel vectors for $\mathcal{U}$.

Theorem 4.3. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system on a finitely or countably generated Hilbert $\mathcal{A}$-module $\mathcal{H}$ over a unital commutative $C^{\ast}$-algebra $\mathcal{A}$. If $\eta\in \mathcal{H}$ is a complete Parseval frame vector, then

Proof. According to Lemma 3.1, we can let $\mathcal{U} = \{L_{U}|_{{\rm ran}P}: U\in \mathcal{U}\}$ and $\eta = P\chi_{I}$, where $P$ is an orthogonal projection in the commutant of $\{L_{U}:U\in \mathcal{U}\}$. Let $\mathcal{L} = \{L_{U}:U\in \mathcal{U}\}''$.

Let's assume $A\in \mathcal{U}''$. Then $A = PTP$ for some $T\in \mathcal{L}$. Thus

Then we have

Therefore, $A\eta \in B_{\mathcal{U}}$.

Conversely, let $\xi\in {\rm ran}P$ be a complete Bessel vector for $\mathcal{U}$. Then, $T\chi_{U} = L_{U}\xi$ defines an adjointable operator $T$. For each $V\in \mathcal{U}$, we have

Therefore, $T\in \mathcal{L}'$. In particular, $T\chi_{I} = \xi$. Let $A = P\pi^{-1}(T^{\ast})P\in \mathcal{U}''$. Note that $\pi^{-1}(T^{\ast})\in \mathcal{L}$ and

Thus, we have

Hence $B_{\mathcal{U}} = \{A\eta: A\in \mathcal{U}^{''}\}$. □

Similar to the complete Bessel vectors, we have the following corollary.

Corollary 4.4. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system on a finitely or countably generated Hilbert $\mathcal{A}$-module $\mathcal{H}$ over a unital commutative $C^{\ast}$-algebra $\mathcal{A}$. Suppose vector $\xi$ is in $\mathcal{H}$, and vector $\eta\in \mathcal{H}$ is a complete Parseval frame vector for $\mathcal{U}$. If there's a unitary operator $A\in \mathcal{U}''$ that makes $\xi = A\eta$, then $\xi$ is also a complete Parseval frame vector for $\mathcal{U}$.

Proof. Through Lemma 3.1, we can assume that $\mathcal{U} = \{L_{U}|_{{\rm ran}P}: U\in \mathcal{U}\}$ and $\eta = P\chi_{I}$, where $P$ is an orthogonal projection in $\{L_{U}:U\in \mathcal{U}\}'$. Let $\mathcal{L} = \{L_{U}:U\in \mathcal{U}\}''$.

First, we assume that there is a unitary operator $A\in \mathcal{U}''$ such that $\xi = A\eta$. Our research reveals that the vector $A\eta$ is a complete Parseval frame vector for $\mathcal{U}$. For $z\in {\rm ran}P$,

The conclusion is obtained. □

5.   Frame multiplicity of $\mathcal{A}$-group-like unitary systems

Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system. Two complete Parseval frame vectors $\xi$ and $\eta$ for $\mathcal{U}$ are said to be equivalent if $\{U\xi\}_{U\in \mathcal{U}}$ and $\{U\eta\}_{U\in \mathcal{U}}$ are unitarily equivalent frames, i.e. there is a unitary operator $W$ such that $WU\xi = U\eta$ for any $U\in \mathcal{U}$. Well, we have the following equivalent condition.

Proposition 5.1. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system. Suppose that $\xi_{1}$ and $\xi_{2}$ are two complete Parseval frame vectors for $\mathcal{U}$, respectively. Then $\{U\xi_{1}\}_{U\in \mathcal{U}}$ and $\{U\xi_{2}\}_{U\in \mathcal{U}}$ are unitarily equivalent frames if and only if there is a unitary operator $\mathcal{W}\in \mathcal{U}'$ such that $\mathcal{W}\xi_{1} = \xi_{2}$.

Proof. Suppose that $\{U\xi_{1}\}_{U\in \mathcal{U}}$ and $\{U\xi_{2}\}_{U\in \mathcal{U}}$ are unitarily equivalent. Then there is a unitary operator $\mathcal{W}$ such that $\mathcal{W}U\xi_{1} = U\xi_{2}$ for all $U\in \mathcal{U}$. In particular, $\mathcal{W}\xi_{1} = \xi_{2}$. Then for any $U, V\in \mathcal{U}$, we have

Then $\mathcal{W}\in \mathcal{U}'$.

Conversely, suppose that there is a unitary operator $\mathcal{W}\in \mathcal{U}'$ such that $\mathcal{W}\xi_{1} = \xi_{2}$. Then $\mathcal{W}U = U\mathcal{W}$ and $\mathcal{W}U\xi_{1} = U\mathcal{W}\xi_{1} = U\xi_{2}$. Therefore, $\{U\xi_{1}\}_{U\in \mathcal{U}}$ and $\{U\xi_{2}\}_{U\in \mathcal{U}}$ are unitarily equivalent frames.□

We use $\mathcal{H}^{(n)}$ to denote $\mathcal{H}\oplus \mathcal{H}\oplus \cdots \oplus \mathcal{H}$ (n copies of $\mathcal{H}$) and $\mathcal{U}^{(n)}$ to denote the set $\{U\oplus U\oplus \cdots \oplus U: U\in \mathcal{U}\}$. It is easy to see $\mathcal{U}^{(n)}$ is an $\mathcal{A}$-group-like unitary system on $\mathcal{H}^{(n)}$. An $\mathcal{A}$-group-like unitary system $\mathcal{U}$ which satisfies $\mathcal{W}(\mathcal{U})\neq \emptyset$ is said to have frame multiplicity $n$ if $n$ is the supremum of all the $k\in \mathbb{N}$ with the property that there exist complete Parseval frame vectors $\eta_{i}(i = 1, 2, \ldots k)$ such that $\eta_{1}\oplus \eta_{2}\oplus\cdots \oplus \eta_{k}$ is a complete Parseval frame vector for $\mathcal{U}^{(k)}$. We need the following to prove that the multiplicity of frame is finite.

Theorem 5.2. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system such that $\mathcal{W}(\mathcal{U})\neq\emptyset$. Then the frame multiplicity of $\mathcal{U}$ is $n$ if and only if there are isometry operators $V_{i}\in {\rm End}_{\mathcal{A}}^{\ast}(\mathcal{H})$ such that $V^{\ast}_{i}\in \mathcal{U}'$ and $V_{i}\mathcal{H}\bot V_{j}\mathcal{H}$ when $i\neq j, i, j = 1, 2, \cdots, n$.

Proof. We only consider the case of $n = 2$. Assume that $\eta_{1}, \eta_{2}\in \mathcal{H}$ are complete Parseval frame vectors for $\mathcal{U}$ such that $\{U\eta_{1}\oplus U\eta_{2}:U\in \mathcal{U}\}$ is a Parseval frame for $\mathcal{H}^{(2)}$. Let $\psi\in \mathcal{W}(\mathcal{U})$ and $V_{i}$ be the analysis operator associated with $\eta_{i}, , i = 1, 2$, respectively. Then, $V_{i}$ is an isometry, and

which implies that $V^{\ast}_{i}\in \mathcal{U}'$.

Since $\eta_{1}, \eta_{2}\in \mathcal{H}$ are complete Parseval frame vectors for $\mathcal{U}$, respectively. Then for $y, z\in \mathcal{H}$,

and

which implies that $\sum_{U\in \mathcal{U}}\langle z, U\eta_{2}\rangle U\eta_{1} = \sum_{U\in \mathcal{U}}\langle y, U\eta_{1}\rangle U\eta_{2} = 0$. Then

Since $V_{1}$, $V_{2}$ have closed ranges, we have $V_{1}\mathcal{H}\bot V_{2}\mathcal{H}$.

Conversely, let $\psi\in \mathcal{W}(\mathcal{U})$ and let $\eta_{i} = V^{\ast}_{i}\psi$. Then $\eta_{i}$ is a complete Parseval frame vector for $\mathcal{U}$. We can identify $V_{i}$ as an analysis operator associated with $\eta_{i}$. Since $V_{1}\mathcal{H}\bot V_{2}\mathcal{H}$ and $V_{1}, V_{2}$ are isometry operators. For any $y, z\in \mathcal{H}$, we have

Therefore $\eta_{1}\oplus \eta_{2}$ is a complete Parseval frame vector for $\mathcal{U}^{(2)}$. □

Below is an essential theorem in this section, which tells us that the frame multiplicity is always finite.

Theorem 5.3. Let $\mathcal{U}$ be an $\mathcal{A}$-group-like unitary system such that $\mathcal{W}(\mathcal{U})\neq\emptyset$. Then the frame multiplicity of $\mathcal{U}$ is finite.

Proof. Instead, suppose the frame multiplicity of $\mathcal{U}$ is infinity. For any $k\in \mathbb{N}$, there exist complete Parseval frame vectors $\eta_{i}(i = 1, 2\ldots k)$ for $\mathcal{U}$ such that $\eta_{1}\oplus \eta_{2}\oplus\cdots \oplus \eta_{k}$ is a complete Parseval frame vector for $\mathcal{U}^{(k)}$. Let $T_{\eta_{i}}$ be the analysis operator associated with $\eta_{i}$, and $P_{i}$ be the orthogonal projection from $\mathcal{H}$ onto $T_{\eta_{i}}(\mathcal{H})$. Just like the evidence presented in Theorems 3.2 and 5.2, we have $P_{i}\in \mathcal{U}'$ and $T_{\eta_{i}}(\mathcal{H})\bot T_{\eta_{j}}(\mathcal{H})$ when $i\neq j$. Since $T_{\eta_{i}}(\mathcal{H}) = P_{i}(\mathcal{H})$, we have $P_{i}(\mathcal{H})\bot P_{j}(\mathcal{H})$ when $i\neq j$. Then $P_{i}P_{j} = P_{j}P_{i} = 0$. Let $Q = \sum^{k}_{i}P_{i}$. It is an orthogonal projection and $Q\leq I$. Let $\psi\in \mathcal{W}(\mathcal{U})$. We have

Since $\langle\eta_{i}, \eta_{i}\rangle\in \mathcal{A}$ is positive element, which lead to a contradiction if we let $k\rightarrow \infty$. □

6.   Conclusions

In this paper, we have introduced the concept of $\mathcal{A}$-group-like unitary system $\mathcal{U}$ and have proved that a complete Parseval frame vector for $\mathcal{U}$ on Hilbert $C^{\ast}$-module can be dilated to a complete wandering vector. Moreover, we have provided the parameterization of complete Bessel vector for $\mathcal{U}$. We also have proved that the frame multiplicity of $\mathcal{U}$ is always finite.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by National Natural Science Foundation of China (No.11671201).

Conflict of interest

We declare that there are no conflicts of interest.