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December 2021 Refinements of norm and numerical radius inequalities
Pintu Bhunia, Kallol Paul
Rocky Mountain J. Math. 51(6): 1953-1965 (December 2021). DOI: 10.1216/rmj.2021.51.1953

## Abstract

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then

${1 \over 4}\left\| {{A^*}A + A{A^*}} \right\| \le {1 \over 8}\left( {{{\left\| {A + {A^*}} \right\|}^2} + {{\left\| {A - {A^*}} \right\|}^2} + {c^2}(A + {A^*}) + {c^2}(A - {A^*})} \right) \le {w^2}(A)$

and

${1 \over 2}\left\| {{A^ * }A + A{A^ * }} \right\| - {1 \over 4}{\left\| {{{(A + {A^ * })}^2}{{(A - {A^ * })}^2}} \right\|^{{1 \over 2}}} \le {w^2}(A) \le {1 \over 2}\left\| {{A^ * }A + A{A^ * }} \right\|,$

where $\left\| \cdot \right\|$, $w( \cdot )$ and $c( \cdot )$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then

$\left\| {A{D^ * }} \right\| \le {\left\| {\int_0^1 {{{\left( {(1 - t)({{\left| A \right|}^2} + {{\left| D \right|}^2})/2 + t\left\| {A{D^ * }} \right\|I} \right)}^2}dt} } \right\|^{{1 \over 2}}} \le {1 \over 2}{\left\| {{{\left| A \right|}^2} + D} \right\|^2},$

where $|A{|^2} = {A^ * }A$ and $|D{|^2} = {D^ * }D$. This is a refinement of a well-known inequality obtained by Bhatia and Kittaneh.

## Citation

Pintu Bhunia. Kallol Paul. "Refinements of norm and numerical radius inequalities." Rocky Mountain J. Math. 51 (6) 1953 - 1965, December 2021. https://doi.org/10.1216/rmj.2021.51.1953

## Information

Received: 12 November 2020; Revised: 4 April 2021; Accepted: 6 April 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.1953

Subjects:
Primary: 47A12
Secondary: 47A30

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