# Some numerical radius inequalities for semi-Hilbert space operators

Research paper by **Kais Feki**

Indexed on: **03 Jan '20**Published on: **02 Jan '20**Published in: **arXiv - Mathematics - Functional Analysis**

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#### Abstract

Let $A$ be a positive bounded linear operator acting on a complex Hilbert
space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. 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-Hilbertian space
$\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ respectively, where
${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for all $x,
y\in\mathcal{H}$. In this paper, we prove 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
$T^{\sharp_A}$ is denoted to be a distinguished $A$-adjoint operator of $T$.
Moreover, some $A$-numerical radius inequalities for products and commutators
of semi-Hilbertian space operators are also obtained.