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

Authors : Kais FEKİ

Pages : 795-810

Doi:10.15672/hujms.730574

View : 17 | Download : 5

Publication Date : 2021-06-07

Article Type : Research Paper

Abstract :For a given bounded positive insert ignore into journalissuearticles values(semidefinite); linear operator $A$ on a complex Hilbert space $\biginsert ignore into journalissuearticles values(\mathcal{H}, \langle \cdot, \cdot\rangle \big);$, we consider the semi-Hilbertian space $\biginsert ignore into journalissuearticles values(\mathcal{H}, \langle \cdot, \cdot\rangle_A \big);$ where ${\langle x, y\rangle}_A := \langle Ax, y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \[\omega_Ainsert ignore into journalissuearticles values(T);=\sup\Big\{\big|{\langle Tx, x\rangle}_A\big|\,;\,\, x\in\mathcal{H},\, {\langle x, x\rangle}_A=1\Big\}.\] 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 $\mathbb{A}=\text{diag}insert ignore into journalissuearticles values(A,A);$.
Keywords : positive operator, operator matrix, semi inner product, A numerical radius, inequality