Abstract
Let A and B be \(n\times n\) positive semidefinite matrices, and let \( ||\cdot ||_{2}\) be the Hilbert-Schmidt norm. Bhatia and then Hayajneh and Kittaneh, using different techniques, proved that
for \(v\in \left[ \frac{1}{4},\frac{3}{4}\right] \), which gives an affirmative answer to an open problem posed by Bourin for the special case of the Hilbert–Schmidt norm. In this paper, we prove a general unitarily invariant norm inequality from which we obtain a new proof of the above Hilbert–Schmidt norm inequality. We also prove that if \(r\ge 1,\) then
for \(\frac{1}{2r}\le v\le \frac{2r-1}{2r}\), where \(|||\cdot |||\) denotes any unitarily invariant norm.
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1 Introduction
For any \(n\times n\) complex matrix X, the eigenvalues and the singular values of X are denoted by \(\lambda _{i}(X)\) and \(\sigma _{i}(X)\) for \( i=1,2,\ldots ,n,\) and arranged in such a way that \(\left| \lambda _{1}(X)\right| \ge \left| \lambda _{2}(X)\right| \ge \cdots \ge \left| \lambda _{n}(X)\right| \) and \(\sigma _{1}(X)\ge \sigma _{2}(X)\ge \cdots \ge \sigma _{n}(X).\) Thus, \(\sigma _{i}(X)=\lambda _{i}(\left| X\right| )\) for \(i=1,2,...,n\), where \(\left| X\right| =(X^{*}X)^{1/2}\) is the absolute value of X.
Recall that for any two \(n\times n\) complex matrices X and Y, we have \( \lambda _{i}(XY)=\lambda _{i}(YX)\) for \(i=1,2,\ldots ,n,\) and \(\left\| X\right\| _{2}=\left( \underset{i=1}{\overset{n}{\sum }}\sigma _{i}^{2}(X)\right) ^{1/2}=\left( \textrm{tr}\left| X\right| ^{2}\right) ^{1/2}\) and \(\left\| X\right\| _{1}=\underset{i=1}{\overset{n}{\sum }}\sigma _{i}(X)=\textrm{tr}\left| X\right| \) are the Hilbert–Schmidt norm and the trace norm of X, respectively. Moreover, any unitarily invariant norm is an increasing function of singular values.
Let A and B be positive semidefinite matrices, and let \(|||\cdot |||\) be any unitarily invariant norm. Bourin [5], in his paper on the subadditivity of concave functions of positive semidefinite matrices, asked whether the inequality
is true.
In their works on the aforesaid conjecture, Bhatia [4] and Hayajneh and Kittaneh [9] proved that
is true whenever \(v\in \left[ \frac{1}{4},\frac{3}{4}\right] \).
A complete answer to Bourin\(^{,}\)s question for the trace norm \(\left\| \cdot \right\| _{1}\) has been given by Hayajneh and Kittaneh [7], that is,
is true for \(v\in \left[ 0,1\right] .\) Several partial solutions to Bourin\( ^{,}\)s problem have been given in [10] and references therein.
In this paper, we prove that if f is a nonnegative concave function on \( \left[ 0,\infty \right) ,\) then
for \(v\in \left[ 0,1\right] \). We also prove the following inequality related to the inequality (1.1)
for \(\frac{1}{2r}\le v\le \frac{2r-1}{2r},\) \(r\ge 1\).
2 Main results
The following lemmas are required in order to support the main results.
Lemma 2.1
[12] Given any positive semidefinite block matrix \( \begin{bmatrix} M &{} K \\ K^{*} &{} N \end{bmatrix} \), where M and N are \(m\times m\) and \(n\times n\) complex matrices, respectively, we have
for \(i=1,\ldots ,r\) and \(r=\min \{m,n\}\).
Lemma 2.2
[3, p. 291] Let X be an \(n\times n\) complex matrix, and let f be a nonnegative increasing function on \(\left[ 0,\infty \right) .\) Then
for \(i=1,2,...,n.\)
Lemma 2.3
[6] Let X, Y, and Z be \(n\times n\) complex matrices such that the block matrix \( \begin{bmatrix} X &{} Y \\ Y^{*} &{} Z \end{bmatrix} \) is positive semidefinite, and let f be a nonnegative concave function on \(\left[ 0,\infty \right) .\) Then
In particular,
Using Lemma 2.1, Lemma 2.2, and Lemma 2.3, we prove our first main result.
Theorem 2.4
Let A and B be positive semidefinite matrices, and let f be a nonnegative concave function on \(\left[ 0,\infty \right) .\) Then
for \(v\in \left[ 0,1\right] .\)
Proof
Let \(r\ge 0,\) and let \(X= \begin{bmatrix} A^{\frac{r}{2}} &{} B^{\frac{r}{2}} \\ B^{\frac{r}{2}} &{} A^{\frac{r}{2}} \end{bmatrix} \) and \(Y= \begin{bmatrix} A &{} 0 \\ 0 &{} B \end{bmatrix} \). Then
is positive semidefinite, and hence by using Lemma 2.1, we have
for \(i=1,2,...,n.\)
Now, for \(i=1,2,...,n,\) we have
So,
Replacing A, B by \(A^{\frac{1}{r+1}},B^{\frac{1}{r+1}},\) respectively, and taking \(v=\frac{r}{r+1}\), we obtain
for \(v\in [0,1]\). Here, we have used the fact that \(\left| \left| \left| f\left( \left| X\right| \right) \right| \right| \right| =\left| \left| \left| f\left( \left| X^{*}\right| \right) \right| \right| \right| \) for any complex matrix X. This completes the proof of the theorem.
Taking \(v=1\) in the aforementioned Theorem 2.4, we have the following corollary.
Corollary 2.5
Let A and B be positive semidefinite matrices, and let f be a nonnegative concave function on \(\left[ 0,\infty \right) .\) Then
As an application of Theorem 2.4, we give a different solution to Bourin\(^{,}\)s question for the Hilbert–Schmidt norm. To achieve this, we need the following lemmas.
Lemma 2.6
[9] Let A and B be positive semidefinite matrices, and let \(v\in \left[ 0,1\right] \). Then
Lemma 2.7
[9] Let A and B be positive semidefinite matrices, and let \(v\in \left[ \frac{1}{2},1\right] \). Then
An equivalent form of the following lemma has been given in [9]. For the reader\(^{,}\)s convenience, we give a short proof of this lemma based on Lemma 2.6 and Lemma 2.7.
Lemma 2.8
Let A and B be positive semidefinite matrices, and let \( v\in \left[ \frac{1}{2},1\right] \). Then
Proof
We can easily check that the square of the left hand side of the inequality ( 2.2) is equal to
Hence, the inequality (2.2) is equivalent to the inequality
In view of Lemma 2.8 and Theorem 2.4, applied to the Hilbert–Schmidt norm and the case \(f(t)=t,\) we have the following corollary.
Corollary 2.9
Let A and B be positive semidefinite matrices, and let \( v\in \left[ \frac{1}{2},1\right] \). Then
It should be mentioned here that Corollary 2.9 can be concluded from Theorem 2.7 in [8], using a completely different analysis.
Corollary 2.10
Let A and B be positive semidefinite matrices, and let \(v\in \left[ \frac{1}{4},\frac{3}{4}\right] \). Then
Proof
Using Corollary 2.9, we have
for \(v\in \left[ \frac{1}{2},1\right] \). Hence, the inequality (2.3) is valid for \(v\in \left[ \frac{1}{4},\frac{1}{2}\right] \). Therefore,
is also valid for \(1-v\in \left[ \frac{1}{4},\frac{1}{2}\right] \), i.e., \( v\in \left[ \frac{1}{2},\frac{3}{4}\right] \). Hence, the inequality (2.3) is valid for \(v\in \left[ \frac{1}{4},\frac{3}{4}\right] \).
To prove our second main result in this paper, we need the following lemmas.
Lemma 2.11
(Matrix Young Inequality) [1] Let A and B be \(n\times n\) complex matrices. Then
for \(i=1,2,...,n,\) and \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\).
Lemma 2.12
[2] Let A and B be positive semidefinite matrices. Then
and
Theorem 2.13
Let A and B be positive semidefinite matrices, and let \(r\ge 1\). Then
for \(\frac{1}{2r}\le v\le \frac{2r-1}{2r}.\)
Proof
Let \(X= \begin{bmatrix} A^{v} &{} B^{v} \\ 0 &{} 0 \end{bmatrix} \) and \(Y= \begin{bmatrix} B^{1-v} &{} A^{1-v} \\ 0 &{} 0 \end{bmatrix} \). Then
and
where the first inequality follows from Lemma 2.11, the second inequality follows from the triangle inequality, and the last equality follows using the fact that \(|||~|X|^{r}|||=|||~|X^{*}|^{r}|||\) for any complex matrix X and for \(r>0\). Hence,
Assume that \(\frac{1}{2}\le v\le \frac{2r-1}{2r}\). Since \(\frac{1}{2v}\le 1\), it follows, by Lemma 2.12, that
It is known [3, p. 95] that \(|||X|||_{(r)}:=|||~|X|^{r}|||^{1/r}\) is a unitarily invariant norm for \(r\ge 1\). Hence, again using Lemma 2.12, we have
Hence, when \(\frac{1}{2}\le v\le \frac{2r-1}{2r},\) we obtain
Moreover, by Lemma 2.12, we have \(|||A+B|||\le |||(A^{1/r}+B^{1/r})^{r}|||\) and this gives the following inequality
Similarly, when \(\frac{1}{2r}\le v\le \frac{1}{2}\), i.e., \(\frac{1}{2}\le 1-v\le \frac{2r-1}{2r}\), we again have the inequality (2.4). Therefore,
for \(\frac{1}{2r}\le v\le \frac{2r-1}{2r}\).
We conclude the paper with the following remark.
Remark 2.14
The case \(v=\frac{1}{2}\) and \(r=1\) in Theorem 2.13 is the inequality
which can also be concluded from the triangle inequality and the arithmetic–geometric mean inequality for unitarily invariant norms (see, e.g., [11]).
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Kittaneh, F., Matharu, J.S. Further norm inequalities for positive semidefinite matrices. J. Appl. Math. Comput. 70, 3281–3289 (2024). https://doi.org/10.1007/s12190-024-02097-1
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DOI: https://doi.org/10.1007/s12190-024-02097-1