q-Numerical radius inequalities for Hilbert space

Abstract

The aim of this paper is to study the q-numerical radius ${\omega }_q(.)$ of bounded linear operators on Hilbert spaces. More precisely, first, we show that ${\omega }_q(.)$ defines a norm which is equivalent to the operator norm. Next, the following compatible generalization of Kittaneh's inequality $\begin{array}{rl}\frac{1}{4}{\left(\frac{q}{2-q^2}\right)}^2\parallel T^{\ast }T+T{T}^{\ast }\parallel & \le {\omega }_q^2(T)\le \frac{(q+2\sqrt{1-q^2}{)}^2}{2}\\ & \times \Vert T^{\ast }T+T{T}^{\ast }\Vert .\end{array}$is obtained. Finally, some generalizations of q-numerical radius inequalities for composition of operators are established.

Keywords:

• Numerical range
• numerical radius

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No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.