Singular value inequalities with applications

Volume 24, Issue 4, pp 323--329 http://dx.doi.org/10.22436/jmcs.024.04.04

Publication Date: April 08, 2021 Submission Date: January 21, 2021 Revision Date: February 18, 2021 Accteptance Date: March 18, 2021

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Authors

Wasim Audeh - Department of Mathematics, University of Petra, Amman, Jordan.

Abstract

Let \(A_{i},B_{i},X_{i},Y_{i}\) be \(n\times n\) complex matrices, \(i=1,2,...,m\) and let \(f\) be a nonnegative increasing convex function on an interval \(I\) such that \(0\in I\) and \(f(0)\leq 0\). Then% \[ 2s_{j}\left( f\left( \left \vert \sum \limits_{i=1}^{m}A_{i}X_{i}Y_{i}^{\ast }B_{i}^{\ast }\right \vert \right) \right) \leq \left( \max \left \{ S,T\right \} \right) ^{2}s_{j}(K) \] for $j=1,2,...,n,$ where% \[ S=\left \Vert \sum \limits_{i=1}^{m}A_{i}A_{i}^{\ast }\right \Vert ^{1/2}\text{% , }T=\left \Vert \sum \limits_{i=1}^{m}B_{i}B_{i}^{\ast }\right \Vert ^{1/2}% \text{,} \] \[ K=f(\left \vert X_{1}\right \vert ^{2}+\left \vert Y_{1}\right \vert ^{2})\oplus ...\oplus f(\left \vert X_{m}\right \vert ^{2}+\left \vert Y_{m}\right \vert ^{2}) \] and \(\max \left \{ S,T\right \} \leq 1\). Several singular value inequalities are also proved.

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Keywords

• Singular value
• convex function
• positive operator
• inequality

MSC

•  15A18
•  15A42
•  15A60
•  47A63
•  47B05
•  47B15

References

• [1] T. Ando, Majorizations and inequalities in matrix theory, Linear Algebra Appl., 199 (1994), 17--67
  • View Article
  • MathSciNet


• [2] W. Audeh, Singular value inequalities and applications, Positivity, 2020 (2020), 10 pages
  • View Article
  • MathSciNet


• [3] W. Audeh, Generalizations for singular value and arithmetic-geometric mean inequalities of operators, J. Math. Anal. Appl., 2020 (489), 8 pages
  • View Article
  • MathSciNet


• [4] W. Audeh, Generalizations for singular value inequalities of operators, Adv. Oper. Theory, 5 (2020), 371--381
  • View Article
  • MathSciNet


• [5] W. Audeh, Some generalizations for singular value inequalities of compact operators, Adv. Oper. Theory, 6 (2021), 10 pages
  • View Article
  • MathSciNet


• [6] W. Audeh, F. Kittaneh, Singular value inequalities for compact operators, Linear Algebra Appl., 437 (2012), 2516--2522
  • View Article
  • MathSciNet


• [7] J. S. Aujla, F. C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl., 369 (2003), 217--233
  • View Article
  • MathSciNet


• [8] R. Bhatia, Matrix Analysis, Springer, New York (1997)
  • View Article
  • MathSciNet


• [9] R. Bhatia, F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl.,, 11 (1990), 272--277
  • View Article
  • MathSciNet


• [10] I. C. Gohberg, M. G. Kreın, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence (1969)
  • View Article
  • MathSciNet


• [11] O. Hirzallah, Inequalities for sums and products of operators, Linear Algebra Appl., 407 (2005), 32--42
  • View Article
  • MathSciNet


• [12] R. A. Horn, C. R. Johnson, Topics in in Matrix Analysis, Cambridge University Press, Cambridge (1991)
  • View Article
  • MathSciNet