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

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, and A be a positive bounded linear operator on $\mathcal{H}.$ Let ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}$ denote the set of all bounded linear operators on $\mathcal{H}$ whose A-adjoint exists. Let $\mathbb{A}$ denote a $2\times 2$ operator matrix of the form $\left[\begin{array}{cc}A& O\\ O& A\end{array}\right]$. 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 $\mathbb{A}$-numerical radius inequalities for operator matrices in ${\mathcal{B}}_{\mathbb{A}}\mathcal{(}\mathcal{H}⨁\mathcal{H}\mathcal{)}$. 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 $\mathbb{A}$-numerical radius of an operator matrix whose entries are operators in ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}.$ Further, we prove some refinements of earlier A-numerical radius inequalities for operators in ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}$.

Keywords:

• A-numerical radius
• positive operator
• semi-inner product
• inequality
• operator matrix

2010 Mathematics Subject Classifications:

• 47A12
• 47A30
• 47A63
• 47A05

Acknowledgments

We thank the referee for the constructive and insightful comments, which have helped us to substantially improve our manuscript. We thank the Government of India for introducing the work from home initiative during the COVID-19 pandemic.

Disclosure statement

No potential conflict of interest was reported by the author(s).