Some bounds for the $\mathbb{A}$-numerical radius of certain $2\times 2$ operator matrices

Abstract

For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $(\mathcal{H},⟨\cdot \mid \cdot ⟩)$, we consider the semi-Hilbertian space $(\mathcal{H},⟨\cdot \mid \cdot {⟩}_A)$ where ${⟨x\mid y⟩}_A:=⟨Ax\mid y⟩$ for every $x,y\in \mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*} ω_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|\,; \,\,x\in \mathcal{H}, \,{\langle x\mid x\rangle}_A= 1\Big\}. \end{align*} Our aim in this paper is to derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are $A$-bounded operators, where $\mathbb{A}=\text{diag}(A,A)$.

It is submitted to a research journal since 1 May 2020