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

Kais Feki
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引用次数: 5

Abstract

For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$, we consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle_A \big)$ where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*} \omega_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 $\mathbb{A}=\text{diag}(A,A)$.
某些$2 \乘以2$算子矩阵的$\mathbb{A}$数值半径的一些界
对于给定的有界正(半定)线性算子 $A$ 在复希尔伯特空间上 $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$,我们考虑半希尔伯特空间 $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle_A \big)$ 在哪里 ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ 对于每一个 $x, y\in\mathcal{H}$. The $A$- an的数值半径 $A$-有界算子 $T$ on $\mathcal{H}$ 是由 \begin{align*} \omega_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*} 本文的目的是推导出几个 $\mathbb{A}$-数值半径不等式 $2\times 2$ 算子矩阵的项是 $A$-有界算子,其中 $\mathbb{A}=\text{diag}(A,A)$.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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