On 𝔸-numerical radius inequalities for 2 × 2 operator matrices

Abstract

Let be a complex Hilbert space, and A be a positive bounded linear operator on Let denote the set of all bounded linear operators on whose A-adjoint exists. Let denote a operator matrix of the form . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several -numerical radius inequalities for operator matrices in . In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption ‘A is strictly positive’. We then establish several new upper and lower bounds for the -numerical radius of an operator matrix whose entries are operators in Further, we prove some refinements of earlier A-numerical radius inequalities for operators in .