# Singular value inequalities with applications

Volume 24, Issue 4, pp 323--329
Publication Date: April 08, 2021 Submission Date: January 21, 2021 Revision Date: February 18, 2021 Accteptance Date: March 18, 2021
• 36 Downloads
• 91 Views

### 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.

### Share and Cite

##### ISRP Style

Wasim Audeh, Singular value inequalities with applications, Journal of Mathematics and Computer Science, 24 (2022), no. 4, 323--329

##### AMA Style

Audeh Wasim, Singular value inequalities with applications. J Math Comput SCI-JM. (2022); 24(4):323--329

##### Chicago/Turabian Style

Audeh, Wasim. "Singular value inequalities with applications." Journal of Mathematics and Computer Science, 24, no. 4 (2022): 323--329

### Keywords

• Singular value
• convex function
• positive operator
• inequality

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

### References

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

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

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

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

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

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

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

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

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

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

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

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