# $A$-numerical radius inequalities for semi-Hilbertian space operators

Research paper by **Ali Zamani**

Indexed on: **13 May '19**Published on: **10 May '19**Published in: **arXiv - Mathematics - Functional Analysis**

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#### Abstract

Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H},
\langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x,
y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm
${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A$ and $w_A(T)$ denote the
$A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in
semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In
this paper, we prove the following characterization of $w_A(T)$ \begin{align*}
w_A(T) = \displaystyle{\sup_{\alpha^2 + \beta^2 = 1}} {\left\|\alpha \frac{T +
T^{\sharp_A}}{2} + \beta \frac{T - T^{\sharp_A}}{2i}\right\|}_A, \end{align*}
where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. We then
apply it to find upper and lower bounds for $w_A(T)$. In particular, we show
that \begin{align*} \frac{1}{2}{\|T\|}_A \leq \max\Big\{\sqrt{1 -
{|\cos|}^2_AT}, \frac{\sqrt{2}}{2}\Big\}w_A(T)\leq w_A(T), \end{align*} where
${|\cos|}_AT$ denotes the $A$-cosine of angle of $T$. Some upper bounds for the
$A$-numerical radius of commutators, anticommutators, and products of
semi-Hilbertian space operators are also given.