Some Singular Value Inequalities for Sector Matrices Involving Operator Concave Functions

Abstract

In this paper, we give some singular value inequalities for sector matrices involving operator concave function, which are generalizations of some existing results. Moreover, we present some unitarily invariant norm inequalities for sector matrices.

1. Introduction

Throughout this paper, let be the set of all complex matrices. We denote by the identity matrix in . For , the conjugate transpose of is denoted by , and the matrices and are called the real part and imaginary part of , respectively (see [1], p. 6 and [2], p. 7). Moreover, is called accretive if . For two Hermitian matrices , we write (resp., ) if is positive semidefinite (resp., positive definite).

Matrix inequalities have many important applications in solving matrix equations; for more information, we refer the readers to see [3, 4]. In this paper, we mainly deal with matrix singular value inequalities.

A real valued continuous function on an interval is called matrix concave of order if for any two Hermitian matrices with spectrum in and all . Furthermore, is called operator concave if is matrix concave for all . We also note that on interval is operator monotone if for any two positive semidefinite with spectra in , implies . It is well-known that is operator monotone if and only if is operator concave.

Let denote any unitarily invariant norm on , which satisfies for any unitary matrices and all . Let denote the singular values of .

The numerical range of is defined by

For , denotes the sector region in the complex plane as follows:

If , then is positive definite, and if for some , then , A is nonsingular, and is positive definite. Moreover, implies for any nonzero matrix , thus . Recent developments on sector matrices can be found in [5–8].

For two positive definite matrices , the geometric mean, harmonic mean, and arithmetic mean are defined, respectively, as follows: , , and . The arithmetic-geometric-harmonic mean inequalities for positive definite matrices state that

For two accretive matrices , Drury [7] defined the geometric mean of and as follows:

This new geometric mean defined by (4) possesses some similar properties compared with the geometric mean of positive matrices. For instance, and . Moreover, if with , then .

For the sake of convenience, we will need the following notation:

For , by a limit argument, we know that . One may wonder what happens if ? We note that . Otherwise, ; since is continuous and increasing, we can find such that , a contradiction. Thus, . Throughout this paper, by function with , we mean and Thus, one can easily verify that is operator monotone.

Recently, Bedrani, Kittaneh, and Sababheh [5] characterized the operator monotone function for an accretive matrix: let be accretive and ,where is the probability measure satisfying .

In [9], Garg and Aujla showed the following inequalities:where and is an operator concave function.

By letting in (7) and (8), we have

Recently, Yang and Lu [10] obtained two singular value inequalities for sector matrices based on (7) and (8). Later, Lin and Fu [11], Xue and Hu [12], and Nasiri and Furuichi [13] independently gave some singular value inequalities for sector matrices related to Yang and Lu’s results.

In this paper, we give some singular value inequalities for sector matrices based on (9) and (10), which are generalizations of some existing results mentioned above. Moreover, we present some unitarily invariant norm inequalities for sector matrices.

2. Main Results

We begin this section with some results from the literature which will be necessary for achieving our goals.

Lemma 2.1. (see [5]). Let with . If , then

Lemma 2.2. (see [1], p.73]). Let be accretive. Then,

Consequently,

Lemma 2.3. (see [8]). Let be such that . Then,

Lemma 2.4. (see [14]). Let be such that for some . Then,

Lemma 2.5. (see [15]). Let be such that for some . Let be the polar decomposition of . Then,

Lemma 2.6. (see [16–18]). Let be such that and . Then,

The inequalities also hold if replacing by .

The following lemma is a well-known result.

Lemma 2.7. (see [19]). If is operator monotone, then for .

Lemma 2.8. (see [20]). If is monotone concave, then for positive semidefinite matrices , there exist unitary matrices such that .

Lemma 2.9. (see [8, 21]). Let with . Then,

Theorem 2.10. Let be such that and let for . Then,where .

Proof. We have the following chain of inequalities:where are unitary matrices.
This completes the proof.

Remark 2.11. Since for some unitary matrix , we haveBy letting in Theorem 2.10, we obtain Theorem 2.8 in [11]; thus, Theorem 2.10 is a generalization of Theorem 2.8 in [11].

Theorem 2.12. Let be such that and let for . Then,where .

Proof. We have the following chain of inequalities:where are unitary matrices.
This completes the proof.

Remark 2.13. By letting in Theorem 2.12 and taking Remark 2.11 into consideration, we note that Theorem 2.12 is a generalization of Theorem 2.6 in [12].

Theorem 2.14. Let be such that . Then,where and .

Proof. For positive semidefinite matrix , we have , which is equivalent to . Thus, we obtainwhich completes the proof.

Theorem 2.15. Let be such that . Then,where .

Proof. By inequality (3), we obtainWe have the following chain of inequalities:which completes the proof.

Example 2.16. LetWe compute the right hand side of (27) and (2.7) in [13], respectively, as follows:This example shows that inequality (27) is stronger than inequality (2.7) in [13].

Example 2.17. LetWe compute the right hand side of (27) and (2.7) in [13], respectively, as follows:This implies that inequality (27) is weaker than inequality (2.7) in [13].

Theorem 2.18. Let be such that and let for . Then,where .

Proof. ComputeTo prove (37), computewhich completes the proof.

Theorem 2.19. Let be such that and let for . Then,

Proof. Let be positive semidefinite. By inequality (6), we obtainBy the property that weak log-majorization implies weak majorization and Cauchy-Schwarz inequality, we getwhich is equivalent toAccording to Fan’s dominance principle (see [1], p.93), we haveHence,Now, we haveTo prove (41), we computewhich completes the proof.

Theorem 2.20. Let be such that . Then,

Proof. We have the following chain of inequalities:which completes the proof.

Remark 2.21. We remark that Theorem 2.20 is a refinement of Proposition 3.2 in [22].

Theorem 2.22. Let be such that and let for . Then,

Proof. We have the following chain of inequalities:which completes the proof.

Remark 2.23. In Theorem 2.22, letting for any , we have a better bound .

We close this paper with a determinant inequality.

Theorem 2.24. Let be such that . Then,

Proof. For positive semidefinite , let and in (5); one can obtainComputecompleting the proof.

Remark 2.25. We remark that Theorem 2.24 is a refinement of Proposition 3.1 in [22].

3. Conclusion

In this paper, we utilize the properties of operator concave functions to obtain some singular value inequalities that generalize some existing results, and some unitarily invariant norm inequalities and determinant inequalities are also derived which are refinements of some known results. In the future, we plan to investigate some related singular value inequalities for sector matrices involving operator convex functions.

Data Availability

No data were used to support the study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This project was funded by China Postdoctoral Science Foundation (No. 2020M681575).