Abstract
It is shown among other inequalities that if A, B and X are \(n\times n\) complex matrices such that A and B are positive semidefinite, then \(s_{j}(AX-XB)\le \) \(s_{j}\left( \left( \frac{1}{2}A+\frac{1}{2}A^{1/2}\left| X^{*}\right| ^{2}A^{1/2}\right) \oplus \left( \frac{1}{2}B+\frac{1}{2} B^{1/2}\left| X\right| ^{2}B^{1/2}\right) \right) \) for \(j=1,2,\ldots ,2n\). Several related singular value inequalities and norm inequalities are also given.
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1 Introduction
Let \( \mathbb {M} _{n}\) be the algebra of all \(n\times n\) complex matrices. For \(A\in \mathbb {M} _{n}\), we denote the eigenvalues of \(\left| A\right| =\left( A^{*}A\right) ^{1/2}\) by \(s_{1}(A)\ge s_{2}(A)\ge \cdots \ge s_{n}(A)\), they are called the singular values of A. Note that \(s_{j}(A)=s_{j}(A^{ *})=s_{j}(\left| A\right| )\) for \(j=1,2,\ldots ,n\). Note that the spectral (usual operator) norm \(\left\| .\right\| \) is the largest singular value, i.e. \(\left\| A\right\| =s_{1}(A)\), and the Schatten p-norms \(\left\| .\right\| _{p}\) are defined interms of the singular values, where \(\left\| A\right\| _{p}=\left( {{\sum _{j=1}^{n} s_{j}^{p}}}(A)\right) ^{1/p}\) for \(1\le p\le \infty \). Apart from the spectral (usual operator) norm and the Schatten p-norms, we have the wider class of unitarily invariant norms \(\left| \left| \left| .\right| \right| \right| \). Unitarily invariant norms are characterized by the invariance property which states that \( \left| \left| \left| UAV\right| \right| \right| =\left| \left| \left| A\right| \right| \right| \) for all \(A\in \mathbb {M} _{n}\) and for all unitary matrices U and V. Unitarily invariant norms are increasing functions of singular values (see, e.g., [4] or [9]).
For \(A,B,X\in \mathbb {M} _{n}\), a matrix of the form \(AX-XA\) is called a commutator, a matrix of the form \(AX-XB\) is called a generalized commutator, a matrix of the form \(AX+XA\) is called anticommutator, and a matrix of the form \(AX+XB\) is called a generalized anticommutator. In this paper, we present singular value inequalities for these types of matrices.
Kittaneh in [11] has proved that if \(A,B\in \mathbb {M} _{n}\) are positive semidefinite, then
for \(j=1,2,\ldots ,2n\). Inequality (1.1) can be extended to unitarily invariant norms. For \(j=1\), this inequality is the spectral norm inequality,
Specifying inequality (1.1) to the Schatten p-norms, we get
for \(1\le p\le \infty .\) Kittaneh in [10] has been proved that if \(A,B\in \mathbb {M} _{n}\) are positive semidefinite, then
It should be mentioned here that inequality (1.4) is trivial consequence of inequality (1.1) by application of triangular inequality. Specifying inequality (1.4) to the spectral norm \( \left\| .\right\| \), leads to
Davidson and Power in [8] has been shown a weaker version of inequality (1.5). Bourin in [7] provides an equivalent formulation of inequality (1.5). Specifying inequality (1.4) to the Schatten p-norms, we have
For recent studies and details for generalizations of singular value inequalities, we refer to [1, 2] and [3]. In this paper, we give a remarkable generalizations of the inequalities (1.1), (1.2), and (1.3). Several applications are also given.
2 Main results
To reach our findings, we need the following lemmas. The first lemma has been shown by Bhatia and Kittaneh in [5]. The second lemma has been proved by Bourin in [6]. The third lemma has been given by Bhatia in [4].
Lemma 2.1
Let \(A,B\in \mathbb {M} _{n}\). Then
for \(j=1,2,\ldots ,n\)
Lemma 2.2
Let \(A,B\in \mathbb {M} _{n}\) be normal and let f be a nonnegative concave function on \(\left[ 0,\infty \right) \). Then
for every unitarily invariant norm.
Lemma 2.3
Let \(A,B\in \mathbb {M} _{n}\) such that AB is Hermitian. Then
From now until the end of the paper, we will assume that all functions considered are continuous and all matrices denoted by the symbol A or B are positive semidefinite. Our first result is the following singular value inequality for generalized commutator.
Theorem 2.4
Let \(A,B,X\in \mathbb {M} _{n}\). Then
for \(j=1,2,\ldots ,2n\), where
and
Proof
Let
and
Then for \(j=1,2,\ldots ,2n\), we have
Our inequality has thus been proved. \(\square \)
Remark 2.5
Letting \(X=I\) in inequality (2.1), we give
for \(j=1,2,\ldots ,2n\). Inequality (2.2) has been proved by Zhan in [12].
Remark 2.6
Letting \(B=A\) in inequality (2.1), we give the following singular value inequality for commutator.
for \(j=1,2,\ldots ,2n\), where
and
We present the following generalization of inequality (1.1), which is singular value inequality for generalized anticommutator.
Theorem 2.7
Let \(A,B,X\in \mathbb {M} _{n}\). Then
for \(j=1,2,\ldots ,2n\), where
and
Proof
Let
and
Then for \(j=1,2,\ldots ,2n\), we have
Inequality (2.3) has thus been substantiated. \(\square \)
Remark 2.8
Letting \(X=I\) in inequality (2.3), we give inequality (1.1). In that sense inequality (2.3) is certainly a generalization of inequality (1.1).
We are now in a position to present our next norm inequality, which is a generalization of the generalized anticommutator.
Theorem 2.9
Let \(A,B,X\in \mathbb {M} _{n}\) and let f be a nonnegative increasing concave function on \(\left[ 0,\infty \right) \). Then
for every unitarily invariant norm, where
and
Proof
Let
and
Then for \(j=1,2,\ldots ,2n\), we have
This implies that,
which is precisely inequality (2.4). \(\square \)
Remark 2.10
Letting \(f(t)=t\) in inequality (2.4), we give norm inequality for generalized anticommutator. In that sense, inequality (2.4) is certainly a generalization of generalized anticommutator norm inequalities.
Specifying inequality (2.4) to the spectral norm and the Schatten p-norms, we give the following norm inequalities for generalized anticommutator which are generalizations of the inequalities (1.2) and (1.3), respectively.
Corollary 2.11
Let \(A,B,X\in \mathbb {M} _{n}\). Then
where
and
Proof
Inequality (2.3) follows by substituting \(f(t)=t\) and by considering the spectral norm in Theorem 2.9. \(\square \)
Remark 2.12
Letting \(X=I\) in Corollary 2.11, we give inequality (1.2).
Corollary 2.13
Let \(A,B,X\in \mathbb {M} _{n}\). Then for \(1\le p\le \infty \), we have
where
and
Proof
Inequality (2.6) follows by substituting \(f(t)=t\) and by considering the Schatten p-norms in Theorem 2.9. \(\square \)
Remark 2.14
Letting \(X=I\) in Corollary 2.13, we give inequality (1.3).
The following two corollaries are applications of Theorem 2.9.
Corollary 2.15
Let \(A,B,X\in \mathbb {M} _{n}\). Then
for every unitarily invariant norm, where
and
Proof
The inequality is an immediate consequence of Theorem 2.9 by letting \(f(t)=\log (t+1)\). \(\square \)
Corollary 2.16
Let \(A,B,X\in \mathbb {M} _{n}\). Then, for \(r\in \left( 0,1\right] \), we have
for every unitarily invariant norm, where
and
Proof
The inequality is an immediate consequence of Theorem 2.9 by letting \(f(t)=t^{r}\) and \(r\in \left( 0,1\right] \). \(\square \)
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Audeh, W. Singular value inequalities and applications. Positivity 25, 843–852 (2021). https://doi.org/10.1007/s11117-020-00790-6
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DOI: https://doi.org/10.1007/s11117-020-00790-6