Abstract
Let A and B be positive semidefinite matrices. For \(t\in \left[ \frac{3}{4},1\right] \) and for every unitarily invariant norm, it is shown that
and for \(t\in \left[ 0,\frac{1}{4}\right] \),
These norm inequalities are sharper than an earlier norm inequality due to Alakhrass and closely related to an open question of Bourin. In fact, they lead to an affirmative solution of Bourin’s question for \(t=\frac{1}{4}\) and \(\frac{3}{4}\), which is a result due to Hayajneh and Kittaneh (Int J Math 32 (2150043):7, 2021).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Throughout this paper, all matrices are assumed to be complex square matrices of the same size. Let A and B be positive semidefinite matrices. For \(t\in \left[ 0,1\right] \), let
In this paper, \({\left| \left| \left| . \right| \right| \right| }\) denotes any unitarily invariant norm on the space of matrices. Among the most important unitarily invariant norms are the usual operator (or the spectral) norm \(\left\| .\right\| _{\infty }\) and the Schatten \(p-\)norm \(\left\| .\right\| _{p}\) for \(1\le p<\infty \).
In [10], and in his work on the subadditivity of concave functions of positive semidefinite matrices, J.C. Bourin asked the following question.
Question 1.1
Let A and B be any two positive semidefinite matrices, and let \(t\in \left[ 0,1\right] \). Is it true that
Question 1.1 has been answered for the Hilbert-Schmidt norm under the condition \(t\in \left[ \frac{1}{4},\frac{3}{4}\right] \), see [5, 19], and [21].
In [13], Hayajneh and Kittaneh proved the following norm inequality for all unitarily invariant norms:
where A and B are positive semidefinite matrices and \(t\in \left[ 0,1\right] \). Consequently,
for \(t\in \left[ 0,1\right] \). This gives an affirmative answer to Question 1.1 for the trace norm.
In [18], the authors proved the following stronger norm inequality for all unitarily invariant norms:
where A and B are positive semidefinite matrices and \(t\in \left[ 0,1\right] \).
They also gave an affirmative answer to Question 1.1 in the case of \(t=\frac{1}{4}\) and \(t=\frac{3}{4}\) for all unitarily invariant norms. Clearly, Question 1.1 is true for \(t=0\) and \(t=1\).
Moreover, a partial answer to this question in the wider class of unitarily invariant norms has been given in [13] by proving that
and
In [1], Alakhrass proved the following norm inequality for all unitarily invariant norms:
where A and B are positive semidefinite matrices and \(t\in \left[ 0,1\right] \).
In [20], and in order to study Question 1.1, the authors proved the following inequality:
where A and B are positive semidefinite matrices, \(t\in \left[ 0,1\right] \), and \(p\ge 1\). In particular,
Note that for the trace norm, the inequality (1.2) is sharper than the inequality (1.1) for all \(t \in [0,1].\) For the Hilbert-Schmidt norm, the inequality (1.2) is sharper than the inequality (1.1) for \(t \in [0,\frac{1}{4}]\) and \(t \in [\frac{3}{4},1].\) However, the inequality (1.1) is sharper than the inequality (1.2) for \(t \in [\frac{1}{4},\frac{3}{4}].\) But we already know that Bourin’s question is true for \(t \in [\frac{1}{4},\frac{3}{4}]\) for the Hilbert-Schmidt norm.
In [12], and in order to try to answer Question 1.1, the authors proved the following inequality involving the spectral radius \(r\left( .\right) \):
where A and B are positive definite matrices and \(t\in \left[ 0,1\right] \).
Consequently, the authors proved that if \(B\le A \le \left( 1+\epsilon \right) ^{2}B\), then
for \(t\in \left[ 0,1\right] ,\epsilon >0\), and for every unitarily invariant norm. Equivalently, if \(\alpha \ge 1\) and if the spectrum of \(AB^{-1}\) lies in the interval \(\left[ 1,\alpha \right] \), then
For a comprehensive account on related trace and norm inequalities, we refer to [2, 3, 7,8,9,10,11,12,13,14,15,16,17,18, 20,21,22, 25,26,28], and references therein.
In Section 2, we introduce a new way of proving the inequality (1.1) without using the notion of majorization.
In Section 2, we will prove the following norm inequalities for all unitarily invariant norms:
where A and B are positive semidefinite matrices and \(t\in \left[ \frac{3}{4},1\right] \), and
where A and B are positive semidefinite matrices and \(t\in \left[ 0,\frac{1}{4}\right] \). These norm inequalities are sharper than the norm inequality (1.1) and closely related to Question 1.1. In fact, they lead to solve Question 1.1 in the case of \(t=\frac{3}{4}\) and \(t=\frac{1}{4}\) for all unitarily invariant norms, which is a result due to Hayajneh and Kittaneh [18].
2 Main Result
We begin with the following lemmas that will be used in proving our main result.
The following lemma can be found in [4, p. 95]. It contains the celebrated Hölder inequality for all unitarily invariant norms.
Lemma 2.1
Let A and B be any two matrices, \(p>1\), and \(\frac{1}{p}+\frac{1}{q}=1\). Then for every unitarily invariant norm, we have
The following lemma, concerning the convexity and concavity of the power functions of positive semidefinite matrices, can be found in [23].
Lemma 2.2
Let A and B be any two positive semidefinite matrices. Then for every unitarily invariant norm, we have
and
The following lemma can be found in [4, p. 258].
Lemma 2.3
Let A and B be any two matrices. Then for every unitarily invariant norm, we have
The following lemma can be found in [4, p. 253].
Lemma 2.4
Let A and B be any two matrices such that the product AB is normal. Then for every unitarily invariant norm, we have
The following lemma can be found in [6]. It contains the celebrated Heinz inequalities for all unitarily invariant norms.
Lemma 2.5
Let A and B be positive semidefinite matrices. Then for \(t\in \left[ 0,1\right] \) and for every unitarily invariant norm, we have
Convention 2.6
For any matrix T and for every unitarily invariant norm, we have
In the following theorem, we introduce a new way of proving the inequality (1.1) without using the notion of majorization.
Theorem 2.7
Let A and B be positive semidefinite matrices, and let \(t\in \left[ 0,1\right] \). Then for every unitarily invariant norm, we have
Proof
For the end points \(t=0\) and \(t=1\), the result is obvious.
Case 1: \(t\in \left( 0,\frac{1}{2}\right] \). In this case, let
For the partition \(\frac{1}{p}+\frac{1}{q}=t+\left( 1-t\right) =1\) and noting that \(\frac{p}{2}=\frac{1}{2t}\ge 1\) and \(\frac{q}{2}=\frac{1}{2\left( 1-t\right) }\le 1\), we have
Case 2: \(t\in \left[ \frac{1}{2},1\right) \). In this case,
This completes the proof. \(\square \)
To prove our main result, we state and prove the following lemma.
Lemma 2.8
Let A and B be positive semidefinite matrices, and let \(t\in \left[ \frac{3}{4},1\right) \). Let
Then for every unitarily invariant norm and for \(p=\frac{1}{2t-1}\) and \(q=\frac{1}{2\left( 1-t\right) }\), we have
Proof
Since \(t\in \left[ \frac{3}{4},1\right) \), it follows that \(\frac{p}{2} \le 1\) and \(\frac{q}{2}\ge 1\). In fact,
and
Now,
Also,
This completes the proof. \(\square \)
Now, we are in a position to state and prove our main result.
Theorem 2.9
Let A and B be positive semidefinite matrices, and let \(t\in \left[ \frac{3}{4},1\right] \). Then for every unitarily invariant norm, we have
Proof
The result is obvious for \(t=1\). Now, for \(t\in \left[ \frac{3}{4},1\right) \), let
For the partition \(\frac{1}{p}+\frac{1}{q}=\left( 2t-1\right) +2\left( 1-t\right) =1\), noting that \(1\le p \le 2\) and \(2 \le q< \infty \), we have
This completes the proof. \(\square \)
Based on Theorem 2.9, we have the following theorem.
Theorem 2.10
Let A and B be positive semidefinite matrices, and let \(t\in \left[ 0,\frac{1}{4}\right] \). Then for every unitarily invariant norm, we have
Proof
We have
This completes the proof. \(\square \)
Remark 2.11
The inequality (2.1) is sharper than the inequality (1.1) for \(t\in \left[ \frac{3}{4},1\right] \), and the inequality (2.2) is sharper than the inequality (1.1) for \(t\in \left[ 0,\frac{1}{4}\right] \). In fact, for \(t\in \left[ \frac{3}{4},1\right] \), we have \(2\left( t-\frac{3}{4}\right) \le t-\frac{1}{2}\Longleftrightarrow t\le 1\), and for \(t\in \left[ 0,\frac{1}{4}\right] \), we have \(2\left( \frac{1}{4}-t\right) \le \frac{1}{2}-t\Longleftrightarrow 0\le t\).
Remark 2.12
Theorem 2.9 and Theorem 2.10 lead to an affirmative solution of Question 1.1 in the case of \(t=\frac{3}{4}\) and \(t=\frac{1}{4}\) for all unitarily invariant norms, which is a result due to Hayajneh and Kittaneh [18].
Data Availability Statement
Not applicable.
References
Alakhrass, M.: Inequalities related to Heinz mean. Linear Multilinear Algebra 64, 1562–1569 (2016)
Ando, T., Hiai, F., Okubo, K.: Trace inequalities for multiple products of two matrices. Math. Inequal. Appl. 3, 307–318 (2000)
Audenaert, K.: A norm inequality for pairs of commuting positive semidefinite matrices. Electron. J. Linear Algebra 30, 80–84 (2015)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Bhatia, R.: Trace inequalities for products of positive definite matrices. J. Math. Phys. 55(013509), 3 (2014)
Bhatia, R., Davis, C.: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. 14, 132–136 (1993)
Bottazzi, T., Elencwajg, R., Larotonda, G., Varela, A.: Inequalities related to Bourin and Heinz means with a complex parameter. J. Math. Anal. Appl. 426, 765–773 (2015)
Bourin, J.C.: Some inequalities for norms on matrices and operators. Linear Algebra Appl. 292, 139–154 (1999)
Bourin, J.C.: Matrix versions of some classical inequalities. Linear Algebra Appl. 416, 890–907 (2006)
Bourin, J.C.: Matrix subadditivity inequalities and block-matrices. Int. J. Math. 20, 679–691 (2009)
Bourin, J.C., Lee, E.Y.: Matrix inequalities from a two variables functional. Int. J. Math. 27(1650071), 19 (2016)
Darweesh, A., Hayajneh, M., Hayajneh, S., Kittaneh, F.: Norm inequalities for positive definite matrices related to a question of Bourin. Linear Multilinear Algebra (2022). https://doi.org/10.1080/03081087.2022.2091507
Hayajneh, M., Hayajneh, S., Kittaneh, F.: Norm inequalities for positive semidefinite matrices and a question of Bourin. Int. J. Math. 28(1750102), 7 (2017)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: On the Ando-Hiai-Okubo trace inequality. J. Operator Theory 77, 77–86 (2017)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: Remarks on some norm inequalities for positive semidefinite matrices and questions of Bourin. Math. Inequal. Appl. 20, 225–232 (2017)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: Norm inequalities related to the arithmetic-geometric mean inequality for positive semidefinite matrices. Positivity 22, 1311–1324 (2018)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: On some classical trace inequalities and a new Hilbert-Schmidt norm inequality. Math. Inequal. Appl. 21, 1175–1183 (2018)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: Norm inequalities for positive semidefinite matrices and a question of Bourin II. Int. J. Math. 32(2150043), 7 (2021)
Hayajneh, M., Hayajneh, S., Kittaneh, F.: A Hilbert-Schmidt norm Inequality for positive semidefinite matrices related to a question of Bourin. Positivity 26, 9 (2022)
Hayajneh, M., Hayajneh, S., Kittaneh, F., Lebaini, I.: Norm inequalities for positive semidefinite matrices and a question of Bourin III. Positivity 26, 13 (2022)
Hayajneh, S., Kittaneh, F.: Lieb-Thirring trace inequalities and a question of Bourin. J. Math. Phys. 54(033504), 8 (2013)
Hayajneh, S., Kittaneh, F.: Trace inequalities and a question of Bourin. Bull. Austral. Math. Soc. 88, 384–389 (2013)
Hirzallah, O., Kittaneh, F.: Non-commutative Clarkson inequalities for n-tuples of operators. Integr. Equ. Oper. Theory 60, 369–379 (2008)
Kittaneh, F.: A note on the arithmetic-geometric mean inequality for matrices. Linear Algebra Appl. 171, 1–8 (1992)
Hoa, D.T.: An the the the inequality for \(t\)-geometric means. Math. Inequal. Appl. 19, 765–768 (2016)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Lin, M.: Remarks on two recent results of Audenaert. Linear Algebra Appl. 489, 24–29 (2016)
Plevnik, L.: On a matrix trace inequality due to Ando, Hiai and Okubo. Indian J. Pure Appl. Math. 47, 491–500 (2016)
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest.
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hayajneh, M., Hayajneh, S. & Kittaneh, F. Remarks on a Question of Bourin for Positive Semidefinite Matrices. Results Math 78, 88 (2023). https://doi.org/10.1007/s00025-023-01870-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-01870-1