Equality of three numerical radius inequalities

Abstract

For an n-by-n matrix A, let w(A) and parallel to A parallel to denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w(A) <= parallel to A parallel to, are known to hold: w(A)(2) <= (parallel to A parallel to(2) + w(A(2)))/2, w(A) <= (parallel to A parallel to + parallel to A(2)parallel to(1/2))/2, and w(A) <= (parallel to A parallel to + w(Delta(t)(A)))/2 (0 <= t <= 1), where Delta(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. (C) 2018 Elsevier Inc. All rights reserved.