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October 2018 Interpolating inequalities for functions of positive semidefinite matrices
Ahmad Al-Natoor, Omar Hirzallah, Fuad Kittaneh
Banach J. Math. Anal. 12(4): 955-969 (October 2018). DOI: 10.1215/17358787-2018-0008

Abstract

Let A, B be positive semidefinite n×n matrices, and let α(0,1). We show that if f is an increasing submultiplicative function on [0,) with f(0)=0 such that f(t) and f2(t1/2) are convex, then |||f(AB)|||2f4(1(4α(1α))1/4)(|||(αf(A)+(1α)f(B))2|||×|||((1α)f(A)+αf(B))2|||) for every unitarily invariant norm. Moreover, if α[0,1] and X is an n×n matrix with X0, then |||f(AXB)|||2f(X)X|||αf2(A)X+(1α)Xf2(B)||||||(1α)f2(A)X+αXf2(B)||| for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.

Citation

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Ahmad Al-Natoor. Omar Hirzallah. Fuad Kittaneh. "Interpolating inequalities for functions of positive semidefinite matrices." Banach J. Math. Anal. 12 (4) 955 - 969, October 2018. https://doi.org/10.1215/17358787-2018-0008

Information

Received: 23 December 2017; Accepted: 12 March 2018; Published: October 2018
First available in Project Euclid: 10 July 2018

zbMATH: 06946298
MathSciNet: MR3858756
Digital Object Identifier: 10.1215/17358787-2018-0008

Subjects:
Primary: 15A60
Secondary: 15A18‎ , 15A42

Keywords: convex function , positive semidefinite matrix , Singular value , submultiplicative function , ‎unitarily invariant norm

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.12 • No. 4 • October 2018
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