Interpolation unitarily invariant norms inequalities for matrices with applications

1.   Introduction

In this paper, $M_{n}(\mathcal{\mathbb{C}})$ stands for the set of all $n \times n$ complex matrices. A symmetric matrix $A\in M_{n}(\mathcal{\mathbb{C}})$ is positive semidefinite, if for every $x\in \mathcal{\mathbb{C}}^n$, we have $\langle Ax, x\rangle \geq 0$. For $H, K \in M_{n}(\mathcal{\mathbb{C}})$, The block matrix $\left[ {\begin{array}{*{20}{c}} H & 0\\ 0&K \end{array}} \right]$ will be denoted by $H \oplus K$, and it is called the direct sum of $H$ and $K$. We will use ${\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert { \cdot } \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}$ to denote a unitarily invariant matrix norm, which satisfies the property that ${\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {A} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert} = {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {UAV} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}$ for all $A, U, V\in M_{n}(\mathcal{\mathbb{C}})$, where $U$ and $V$ are "unitary matrices" while we will use ${\left\vert\kern-0.1ex\left\vert {\cdot} \right\vert\kern-0.1ex\right\vert}$ to denote the usual operator or spectral norm. For $A\in M_{n}(\mathcal{\mathbb{C}})$, $s_{i}(A)$ will denote $\it{ith}$ largest singular value of $A$, which is the $\it{ith}$ largest eigenvalue of ${\left\vert {A} \right\vert} = \left ({A^*A} \right)^\frac{1}{2}$. It is well known that ${\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {\left[ {\begin{array}{*{20}{c}} 0&H\\ H^* & 0 \end{array}} \right]} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert} = {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {H \oplus H} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}$, and ${\left\vert\kern-0.1ex\left\vert {H \oplus K} \right\vert\kern-0.1ex\right\vert} = \max({\left\vert\kern-0.1ex\left\vert {H} \right\vert\kern-0.1ex\right\vert}, {\left\vert\kern-0.1ex\left\vert {K} \right\vert\kern-0.1ex\right\vert})$ for all $H, K \in M_{n}(\mathcal{\mathbb{C}})$. We refer to ^[7] for more about unitarily invariant matrix norms and singular values.

The arithmetic–geometric mean inequality for matrices, obtained in ^[8], states that if $H, K \in M_{n}(\mathcal{\mathbb{C}})$, then

and the Cauchy–Schwarz inequality states that

The author in ^[4] obtained two main results; in both of them, he used an increasing convex nonnegative function $f$ defined on an interval $I$ that contains the number 0 with $f(0) \leq 0$. The first result states that if $A, B, P$, and $Q \in M_{n}(\mathcal{\mathbb{C}})$ with $\max \left\{ {{\left\vert\kern-0.1ex\left\vert {A} \right\vert\kern-0.1ex\right\vert}, {\left\vert\kern-0.1ex\left\vert {B} \right\vert\kern-0.1ex\right\vert}} \right\} \leq 1$, then

for all $i = 1, 2, \dots, n$. In the second result, he obtained that if $A_j, B_j, P_j$, and $Q_j \in M_{n}(\mathbb{C})$, where $j = 1, 2, \dots, m$, then

for all $i = 1, 2, \dots, n$, where

and

Letting $f(t) = t$, the norm version of inequality (1.3) is given by

while the norm version of inequality (1.4) is given by

where $L$ and $M$ are the same as given formerly, but

For more inequalities related to the inequalities (1.3)–(1.6), we refer to ^[1,5]. And for some inequalities of interpolation type, we refer to ^[10].

In this paper, interpolation inequalities that can be considered generalizations of the inequalities (1.5) and (1.6) are introduced, and many other consequences and applications of these generalizations are also presented.

2.   Results

We begin this section by introducing three lemmas; these lemmas support the proof of the first main result of this paper. The first lemma can be obtained using "The min-max principle" (see, e.g., [7, p. 75]); in addition, it is a direct consequence of a result introduced in [9, p. 27]. The second lemma can be found in [7, p. 253], and the last lemma was introduced in ^[1].

Lemma 2.1. Let $H, K$, and $L \in M_{n}(\mathbb{C})$. Then

Lemma 2.2. Let $P, Q \in M_{n}(\mathbb{C})$ such that $PQ$ is normal. Then

Lemma 2.3. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$. Then

for every real number $c \in [0, 1]$.

It should be mentioned here that the inequality in Lemma 2.3 interpolates between the arithmetic-geometric mean inequality (1.1) $\left({c = \frac{1}{2}, P = Q = I} \right)$ and the Cauchy-Schwarz inequality (1.2) $\left({c = 0\, {\rm{or}}\, 1, P = Q = I} \right)$. For more results on interpolation inequalities, we refer to ^[1,2,3,6].

Now, we will give the first main result in this paper.

Theorem 2.4. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$. Then

for every real number $c \in [0, 1]$.

Proof. Using Lemma 2.3, we get that

Now let $F_{c}$ be the block $2 \times 2$ matrix defined as $F_{c} = \left[ {\begin{array}{*{20}{c}} {\sqrt c P} & 0\\ {\sqrt {1 - c} Q} & 0 \end{array}} \right]$. Then

therefore

Similarly, we can get the following equality:

Combining the inequality (2.2) with the equalities (2.3) and (2.4) leads to

By using Lemmas 2.1 and 2.2, we can get that

So

Similarly

The result follows immediately from the inequality (2.5) and the inequalities (2.6) and (2.7). □

Note that substituting $c = \frac{1}{2}$ in the inequality (2.1) leads us directly to the inequality (1.5), which means that Theorem 2.4 generalizes the inequality (1.5).

Corollary 2.5. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$. Then

Proof. Substitute $c = 0$ or $c = 1$ in the inequality (2.1) to get the result directly. □

Corollary 2.5 illustrates that the inequality (2.1) can be considered an interpolation inequality between the inequalities (1.5) and (2.8).

Corollary 2.6. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$ such that $P$ and $Q$ are positive semidefinite matrices. Then

for every real number $c \in [0, 1]$.

Proof. Replace $P$ and $Q$ in the inequality (2.1) by $P^\frac{1}{2}$ and $Q^\frac{1}{2}$ respectively to get the result directly. □

Now we are ready to give the second main result in this paper.

Theorem 2.7. Let $A_j, B_j, P_j$, and $Q_j \in M_{n}(\mathbb{C})$, where $j = 1, 2, \dots, m$. For a real number $c \in [0, 1]$, we have

where

and

Proof. Consider the following $m \times m$ block matrices

Through simple and direct calculations, we get that

Thus

Also, it is an easy task to see that

and

We get our result by applying the inequality (2.1) to the block matrices $A, B, P,$ and $Q$ and using the Eqs (2.11) to (2.15). □

Note that substituting $c = \frac{1}{2}$ in the inequality (2.10) leads us directly to the inequality (1.6), which means that Theorem 2.7 is a generalization of the inequality (1.6).

Corollary 2.8. Let $A_j, B_j, P_j$, and $Q_j \in M_{n}(\mathbb{C})$, where $j = 1, 2, \dots, m$. Then

where

and

Proof. Substitute $c = 0$ or $c = 1$ in the inequality (2.10) to get the result directly. □

The inequality (2.10) can be viewed as an interpolation inequality between the inequalities (1.6) and (2.16), as demonstrated by Corollary 2.8.

The following corollary is nothing but constraining the inequality (2.10) for two pairs of matrices.

Corollary 2.9. Let $A_j, B_j, P_j$, and $Q_j \in M_{n}(\mathbb{C})$, where $j = 1, 2$. For a real number $c \in [0, 1]$, we have

where

and

Proof. Let $A = P = Q = B = 0$ for $3 \leq j \leq n$ in inequality (2.10) to get the result directly. □

Corollary 2.10. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$, where $P$ and $Q$ are positive semidefinite. For a real number $c \in [0, 1]$, we have

where

Proof. Replace $A_1$ and $B_2$ by $A$, replace $A_2$ and $B_1$ by $B$, and let $P_1 = P_2 = P^\frac{1}{2}$ and $Q_1 = Q_2 = Q^\frac{1}{2}$ in the inequality (2.17) to get the result. □

Now we will present two directly proven inequalities in the subsequent two corollaries.

Corollary 2.11. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$, where $P$ and $Q$ are positive semidefinite. Then

where

Proof. Substitute $c = \frac{1}{2}$ in inequality (2.18) to get the result directly. □

Corollary 2.12. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$, where $P$ and $Q$ are positive semidefinite. Then

Proof. Substitute $c = 0$ or $c = 1$ in the inequality (2.18) to get the result directly. □

It can be observed that the inequality (2.18) is an interpolation inequality between the inequalities (2.19) and (2.20).

Remark 2.13. Substitute $P = Q = X$, where $X \in M_{n}(\mathbb{C})$ is positive semidefinite, in the inequality (2.20) to get that

letting $X = I$ gives the following inequality:

Corollary 2.14. Let $A, B, P$, and $Q \in M_{n}(\mathbb{C})$ be positive semidefinite. Then for a real number $c \in [0, 1]$, we have

where

and

Letting $P = Q = X$. We get that

Proof. Let $A_1 = B_1 = A^\frac{1}{2}$, $A_2 = B_2 = B^\frac{1}{2}$, $P_1 = P_2 = P^\frac{1}{2}$, and $Q_1 = Q_2 = Q^\frac{1}{2}$ in the inequality (2.17) to get the inequality (2.23). □

Corollary 2.15. Let $A, B, P_1, P_2, Q_1, Q_2 \in M_{n}(\mathbb{C})$. For a real number $c \in [0, 1]$, we have

where

Proof. Let $A_1 = B_1 = A$, $A_2 = -B_2 = B$ in the inequality (2.17) to get the inequality (2.25). □

It can be directly deduced that the inequality (2.25) can be considered an interpolation inequality between the inequalities (2.26) and (2.27) that are demonstrated by the subsequent two corollaries.

Corollary 2.16. Let $A, B, P_1, P_2, Q_1, Q_2 \in M_{n}(\mathbb{C})$. Then

where

Proof. Substitute $c = \frac{1}{2}$ in inequality (2.25) to get the result directly. □

Corollary 2.17. Let $A, B, P_1, P_2, Q_1, Q_2 \in M_{n}(\mathbb{C})$. Then

where

Proof. Substitute $c = 0$ or $c = 1$ in the inequality (2.25) to get the result directly. □

Remark 2.18. Substitute $P_2 = Q_2 = B = 0$ in the inequality (2.25) to get that

where

Corollary 2.19. Let $A, B$, and $X \in M_{n}(\mathbb{C})$, where $X$ is positive semidefinite. Then

Proof. In the inequality (2.25), replace $A$ and $B$ by $\left[ {\begin{array}{*{20}{c}} A\\ B \end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}} A\\ -B \end{array}} \right]$ respectively and let $P_1 = P_2 = Q_1 = Q_2 = X^\frac{1}{2}$, to get that the left-hand side of this inequality equals

while the right-hand side of this inequality equals

The result follows directly from the above discussion. □

3.   Conclusions

Wasim Audeh latterly obtained two matrix singular values inequalities. The norm versions of these inequalities are provided in this paper, and we utilize a recent result by Mohammad Al-khlyleh to derive an interpolation inequalities that are related to Audeh's inequalities.

Author contributions

Mohammad Al-Khlyleh: Writing-original draft, Methodology; Mohammad Abdel Aal: Funding acquisition, Supervision; Mohammad F. M. Naser: Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

Authors are grateful to the Middle East University in Amman, Jordan, for the financial support granted to cover the publication fees of this research article.

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.