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

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, \[ \frac14 \|A^*A+AA^*\|\leq w^2(A) \leq \frac12 \|A^*A+AA^*\|. \]