Banach Journal of Mathematical Analysis

Interpolating inequalities for functions of positive semidefinite matrices

Ahmad Al-Natoor, Omar Hirzallah, and Fuad Kittaneh

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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.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 955-969.

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1531209674

Digital Object Identifier
doi:10.1215/17358787-2018-0008

Mathematical Reviews number (MathSciNet)
MR3858756

Zentralblatt MATH identifier
06946298

Subjects
Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 15A42: Inequalities involving eigenvalues and eigenvectors

Keywords
convex function submultiplicative function positive semidefinite matrix singular value unitarily invariant norm

Citation

Al-Natoor, Ahmad; Hirzallah, Omar; Kittaneh, Fuad. Interpolating inequalities for functions of positive semidefinite matrices. Banach J. Math. Anal. 12 (2018), no. 4, 955--969. doi:10.1215/17358787-2018-0008. https://projecteuclid.org/euclid.bjma/1531209674


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