# Singular value and unitarily invariant norm inequalities for sums and products of operators

Singular value and unitarily invariant norm inequalities for sums and products of operators 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}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Operator Theory Springer Journals

# Singular value and unitarily invariant norm inequalities for sums and products of operators

, Volume 6 (4) – Aug 11, 2021
14 pages

/lp/springer-journals/singular-value-and-unitarily-invariant-norm-inequalities-for-sums-and-bqcpmIMn0A
Publisher
Springer Journals
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}.