Singular value inequalities of matrices via increasing functions

Abstract

Abstract Let A, B, X, and Y be n × n $n\times n$ complex matrices such that A is self-adjoint, B ≥ 0 $B\geq 0$ , ± A ≤ B $\pm A\leq B$ , max ( ∥ X ∥ 2 , ∥ Y ∥ 2 ) ≤ 1 $\max ( \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} ) \leq 1$ , and let f be a nonnegative increasing convex function on [ 0 , ∞ ) $[ 0,\infty ) $ satisfying f ( 0 ) = 0 $f(0)=0$ . Then 2 s j ( f ( | X A Y ∗ | ) ) ≤ max { ∥ X ∥ 2 , ∥ Y ∥ 2 } s j ( f ( B + A ) ⊕ f ( B − A ) ) $$ 2s_{j}\bigl(f \bigl( \bigl\vert XAY^{\ast } \bigr\vert \bigr) \bigr)\leq \max \bigl\{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \bigr\} s_{j}\bigl(f(B+A)\oplus f(B-A)\bigr) $$ for j = 1 , 2 , … , n $j=1,2,\ldots,n$ . This singular value inequality extends an inequality of Audeh and Kittaneh. Several generalizations for singular value and norm inequalities of matrices are also given.

Keywords

• Singular value
• Positive semidefinite matrix
• Increasing function
• Convex function
• Unitarily invariant norm