Equality of three numerical radius inequalities

Hwa Long Gau; Pei Yuan Wu

doi:10.1016/j.laa.2018.05.021

Equality of three numerical radius inequalities

Hwa Long Gau, Pei Yuan Wu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

For an n-by-n matrix A, let w(A) and ‖A‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A)≤‖A‖, are known to hold: w(A)²≤(‖A‖²+w(A²))/2, w(A)≤(‖A‖+‖A²‖^1/2)/2, and w(A)≤(‖A‖+w(Δ_t(A)))/2 (0≤t≤1), where Δ_t(A) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities.

Original language	English
Pages (from-to)	51-67
Number of pages	17
Journal	Linear Algebra and Its Applications
Volume	554
DOIs	https://doi.org/10.1016/j.laa.2018.05.021
State	Published - 1 Oct 2018

Keywords

Generalized Aluthge transform
Numerical radius
Numerical range
Operator norm

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AU - Gau, Hwa Long

AU - Wu, Pei Yuan

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N2 - For an n-by-n matrix A, let w(A) and ‖A‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A)≤‖A‖, are known to hold: w(A)2≤(‖A‖2+w(A2))/2, w(A)≤(‖A‖+‖A2‖1/2)/2, and w(A)≤(‖A‖+w(Δt(A)))/2 (0≤t≤1), where Δt(A) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities.

AB - For an n-by-n matrix A, let w(A) and ‖A‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A)≤‖A‖, are known to hold: w(A)2≤(‖A‖2+w(A2))/2, w(A)≤(‖A‖+‖A2‖1/2)/2, and w(A)≤(‖A‖+w(Δt(A)))/2 (0≤t≤1), where Δt(A) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities.

KW - Generalized Aluthge transform

KW - Numerical radius

KW - Numerical range

KW - Operator norm

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DO - 10.1016/j.laa.2018.05.021

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SN - 0024-3795

VL - 554

SP - 51

EP - 67

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Equality of three numerical radius inequalities

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Keywords

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