Some numerical radius inequalities for semi-Hilbert space operators
J. Korean Math. Soc.
Published online July 15, 2021
Kais Feki
University of Sfax
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 [Studia Math. 168 (2005),no. 1, 73-80] that
\begin{equation*}\label{m1}
\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 to compute the $\mathbb{A}$-numerical radius of the operator matrix $\begin{pmatrix}
I&T\\
0&-I
\end{pmatrix}$ where $\mathbb{A}= \text{diag}(A,A)$. In addition, several $A$-numerical radius inequalities for semi-Hilbert space operators are also established.
Keywords : Positive operator, semi-inner product, numerical radius, $A$-adjoint operator, inequality.
MSC numbers : 47A12, 46C05, 47B65, 47A05.
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