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References (20)
-
JC Bourin, M Uchiyama (2007)
A matrix subadditivity inequality for f(A+B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(A+B)$$\end{document} and f(A)+f(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(A)+f(B)$$\end{document}Linear Algebra Appl., 423
-
F Kittaneh (2006)
Norm inequalities for sums of positive operators IIPositivity, 10
-
M Niezgoda (2009)
Commutators and accretive operatorsLinear Algebra Appl., 431
-
O Hirzallah (2016)
Singualr values of convex functions of operators and the arithmetic-geometric mean inequalityJ. Math. Anal. Appl., 433
-
W Audeh (2020)
Singular value inequalities and applicationPositivity
-
R Bhatia (1997)
10.1007/978-1-4612-0653-8Matrix Analysis
-
X Zhan (2000)
Singular values of differences of positive semidefinite matricesSIAM J. Matrix Anal. Appl., 22
-
O Hirzallah (2005)
Inequalities for sums and products of operatorsLinear Algebra Appl., 407
-
R Bhatia, F Kittaneh (2008)
The matrix arithmatic-geometric mean inequality revisitedLinear Algebra Appl., 428
-
F Kittaneh (2007)
Inequalities for commutators of positive operatorsJ. Funct. Anal., 250
-
O Hirzallah, F Kittaneh (2010)
Singular values, norms, and commutatorsLinear Algebra Appl., 432
-
R. Horn, Charles Johnson (1985)
Matrix analysis -
F Kittaneh (1997)
Norm inequalities for certain operator sumsJ. Funct. Anal., 103
-
R Bhatia, F Kittaneh (1990)
On the singular values of a product of operatorsSIAM J. Math. Anal. Appl., 11
-
O Hirzallah, F Kittaneh, K Shebrawi (2011)
Numerical radius inequalities for commutators of Hilbert space operatorsNumer. Funct. Anal. Optim., 32
-
(2005)
Herimitain operators and convex functions -
W Audeh, F Kittaneh (2012)
Singular values inequalities for compact operatorsLinear Algebra Appl., 437
-
M Bakherad, F Kittaneh (2019)
Numerical radius inequalities involving commutators of G1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1}$$\end{document} operatorsComplex Anal. Oper. Theory, 13
-
F Kittaneh (1988)
Notes on some inequalities for Hilbert space operatorsPubl. Res. Inst. Math. Sci., 24
-
(2019)
Positive Definite MatricesOperator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
- Publisher
- Springer Journals
- Copyright
- Copyright © Tusi Mathematical Research Group (TMRG) 2021
- ISSN
- 2662-2009
- eISSN
- 2538-225X
- DOI
- 10.1007/s43036-021-00160-3
- Publisher site
- See Article on Publisher Site
Abstract
In this note, we mainly investigate singular value and unitarily invariant norm inequalities for sums and products of operators. First, we present singular value inequality for the quantity AX+YB\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AX+YB$$\end{document}: let A, B, X and Y∈B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y \in B({\mathcal {H}})$$\end{document} such that both A and B are positive operators. Then sj(AX+YB)⊕0≤sj(K+M)⊕(L1+N),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} s_{j}\left( (AX+YB)\oplus 0\right) \le s_{j}\left( (K+M)\oplus (L_{ 1} + N)\right) , \end{aligned}$$\end{document}for j=1,2,…\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$j = 1,2,\ldots $$\end{document}, where K=12A+12A12|X∗|2A12\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K=\frac{1}{2}A+\frac{1}{2}A^{\frac{1}{2}}|X^{*}|^{2}A^{\frac{1}{2}}$$\end{document}, L1=12B+12B12|Y|2B12\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ L_{1}=\frac{1}{2}B+\frac{1}{2}B^{\frac{1}{2}}|Y|^{2}B^{\frac{1}{2}}$$\end{document}, M=12|B12(X+Y)∗A12|\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=\frac{1}{2}|B^{\frac{1}{2}}(X+Y)^{*}A^{\frac{1}{2}}|$$\end{document} and N=12|A12(X+Y)B12|\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N=\frac{1}{2}|A^{\frac{1}{2}}(X+Y)B^{\frac{1}{2}}|$$\end{document}. In addition, based on the above singular value inequality, we establish a unitarily invariant norm inequality for concave functions. These results generalize inequalities obtained by Audeh directly. Finally, we present another more general singular value inequality for ∑i=1mAi∗Xi∗Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum \nolimits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}$$\end{document}: sj∑i=1mAi∗Xi∗Bi⊕0≤sj∑i=1mAi∗fi2(|Xi|)Ai⊕∑i=1mBi∗gi2(|Xi∗|)Bi,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} s_{j}\left( \sum \limits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}\oplus 0\right) \le s_{j} \left( \left( \sum \limits _{i=1}^{m}A_{i}^{*}f_{i}^{2}(|X_{i}|)A_{i}\right) \oplus \left( \sum \limits _{i=1}^{m}B_{i}^{*}g_{i}^{2}(|X_{i}^{*}|)B_{i}\right) \right) , \end{aligned}$$\end{document}for j=1,2,…\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$j=1,2,\ldots $$\end{document}, where Ai\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{i}$$\end{document}, Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_{i}$$\end{document} and Xi∈B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_{i}\in B({\mathcal {H}})$$\end{document} such that Ai\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{i}$$\end{document} and Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_{i}$$\end{document} (i=1,2,…,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots ,m$$\end{document}) are compact operators and fi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_{i}$$\end{document}, gi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g_{i}$$\end{document} (i=1,2,…,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots ,m$$\end{document}) are 2m nonnegative continuous functions on [0,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[0,+\infty )$$\end{document} with fi(t)gi(t)=t\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_{i}(t)g_{i}(t)=t$$\end{document} (i=1,2,…,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots ,m$$\end{document}) for t∈[0,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in [0,+\infty )$$\end{document}.
Journal
Advances in Operator Theory – Springer Journals
Published: Aug 11, 2021
Keywords: Singular value; Concave functions; Unitarily invariant norms; Schatten p-norms; Commutator; Positive operators; Compact operators; 15A18; 15A60