×

On some singular value inequalities for matrices. (English) Zbl 1322.47018

The authors generalize a result of L.-M. Zou and Y.-Y. Jiang [J. Math. Inequal. 6, No. 2, 279–287 (2012; Zbl 1259.15030)] by showing that, if \(A, B \in M_n\) are positive semidefinite and \(r\) is a positive integer, then \[ 2s_j(A^{1/2}(A + B)^{r-1}B^{1/2}+ A^{1/2}B^{1/2}) \leq s_j((A+B)^r+(A+B))\text{ for }1\leq j \leq n. \] They also present an \(X\)-version of an inequality due to R. Bhatia and F. Kittaneh [Linear Algebra Appl. 428, No. 8–9, 2177–2191 (2008; Zbl 1148.15014)] by proving that, if \(A_1, A_2, B_1, B_2, X \in M_n\), \(f\) and \(g\) are continuous nonnegative functions on \([0,1)\) satisfying the relation \(f(t)g(t) = t\), \(t\in [0,1)\), then \[ s_j(A_1^*XB1 + A_2^*XB_2) \leq s_j((A_1^*f^2(|X^*|)A_1 + A_2^*f^2(|X^*|)A_2)\oplus (B_1^*g^2(|X|)B_1 + B_2^*g^2(|X|)B_2)). \]

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
PDFBibTeX XMLCite