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Inequalities for the norm and numerical radius for Hilbert $C^{*}$-module operators

Mohsen Shah Hosseini^a, Baharak Moosavi<sup>b

a</sup> Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
^b Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran

Abstract: In this paper, we introduce some inequalities between the operator norm and the numerical radius of adjointable operators on Hilbert $C^{*}$-module spaces. Moreover, we establish some new refinements of numerical radius inequalities for Hilbert space operators. More precisely, we prove that if $T \in B(H)$ and
$$ \min \Big( \frac{\Vert T+ T^* \Vert^ 2 }{2}, \frac{\Vert T- T^* \Vert^ 2 }{2}\Big) \leq \max \Big(\inf_{ \Vert x \Vert=1}{\Vert Tx \Vert^2}, \inf_{ \Vert x \Vert=1}\Vert T^*x \Vert^2\Big), $$
then
\begin{equation*} \Vert T \Vert \leq \sqrt{ 2} \omega(T); \end{equation*}
this is a considerable improvement of the classical inequality
\begin{equation*} \Vert T \Vert \leq 2\omega(T). \end{equation*}


Keywords: bounded linear operator, Hilbert space, norm inequality, numerical radius.

UDC: 517.98

MSC: Primary 47A12; Secondary 47A30

Received: 01.12.2019
Revised: 04.06.2020
Accepted: 05.06.2020

Language: English

DOI: 10.15393/j3.art.2020.7330

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