Some numerical radius inequalities for semi-Hilbert space operators
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Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. Let $\omega_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbert space $\big(\mathcal{H}, {\langle \cdot, \cdot\rangle}_A\big)$, respectively, where ${\langle x, y\rangle}_A := \langle Ax, y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show with different techniques from that used by Kittaneh in \cite{FK} that \begin{equation*} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le \omega_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A. \end{equation*} Here $T^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $T$. Moreover, a considerable improvement of the above inequalities is proved. This allows us to compute the $\mathbb{A}$-numerical radius of the operator matrix $\left(\begin{smallmatrix} I&T\\ 0&-I \end{smallmatrix}\right)$ where $\mathbb{A}= \text{diag}(A,A)$. In addition, several $A$-numerical radius inequalities for semi-Hilbert space operators are also established.
Abstract
Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. Let $\omega_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbert space $\big(\mathcal{H}, {\langle \cdot, \cdot\rangle}_A\big)$, respectively, where ${\langle x, y\rangle}_A := \langle Ax, y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show with different techniques from that used by Kittaneh in \cite{FK} that \begin{equation*} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le \omega_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A. \end{equation*} Here $T^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $T$. Moreover, a considerable improvement of the above inequalities is proved. This allows us to compute the $\mathbb{A}$-numerical radius of the operator matrix $\left(\begin{smallmatrix} I&T\\ 0&-I \end{smallmatrix}\right)$ where $\mathbb{A}= \text{diag}(A,A)$. In addition, several $A$-numerical radius inequalities for semi-Hilbert space operators are also established.
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