Rocky Mountain Journal of Mathematics

Reverse Young-type inequalities for matrices and operators

Mojtaba Bakherad, Mario Krnic, and Mohammad Sal Moslehian

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We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal {H})$ are positive operators and $r\geq 0$, $A\nabla _{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp _{-r}B$. We also prove that equality holds if and only if $A=B$. In addition, we establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A$ and $B$ are positive definite matrices and $r\geq 0$, then $\label {reverse_trace} \mbox {tr\,}((1+r)A-rB)\leq \mbox {tr}|A^{1+r}B^{-r} |-r( \sqrt {\mbox {tr\,} A} - \sqrt {\mbox {tr} B})^{2}$.

Article information

Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1089-1105.

First available in Project Euclid: 19 October 2016

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Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 47A30: Norms (inequalities, more than one norm, etc.) 47A60: Functional calculus

Young inequality positive operator operator mean unitarily invariant norm determinant trace


Bakherad, Mojtaba; Krnic, Mario; Moslehian, Mohammad Sal. Reverse Young-type inequalities for matrices and operators. Rocky Mountain J. Math. 46 (2016), no. 4, 1089--1105. doi:10.1216/RMJ-2016-46-4-1089.

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