On 𝔸-numerical radius inequalities for 2 × 2 operator matrices
Nirmal Chandra Rout, Satyajit Sahoo, Debasisha Mishra
Abstract
Let H be a complex Hilbert space, and A be a positive bounded linear operator on H. Let BA(H) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2×2 operator matrix of the form AOOA. 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 A-numerical radius inequalities for operator matrices in BA(H⨁H). 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 A-numerical radius of an operator matrix whose entries are operators in BA(H). Further, we prove some refinements of earlier A-numerical radius inequalities for operators in BA(H).