Euclidean operator radius inequalities of a pair of bounded linear operators and their applications

TL;DR

This paper establishes new sharp bounds for the Euclidean operator radius of pairs of bounded linear operators on complex Hilbert spaces and applies these to improve bounds on the classical numerical radius.

Contribution

It introduces novel bounds for the Euclidean operator radius and derives improved inequalities for the numerical radius of bounded linear operators.

Findings

Derived sharp bounds for Euclidean operator radius.

Improved existing bounds for the numerical radius.

Provided applications to operator inequalities.

Abstract

We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator $A,$ \[\frac{1}{4} \|A^*A+AA^*\|+\frac{\mu}{2}\max \{\|\Re(A)\|,\|\Im(A)\|\} \leq w^2(A) \, \leq \, w^2( |\Re(A)| +i |\Im(A)|),\] where $\mu= \big| \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\|\big|.$ This improve the existing upper and lower bounds of the numerical radius, namely, \[ \frac14 \|A^*A+AA^*\|\leq w^2(A) \leq \frac12 \|A^*A+AA^*\|. \]