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, $$\frac{1}{4} \Vert A^*A+AA^*\Vert +\frac{\mu }{2}\max \{\Vert \Re (A)\Vert ,\Vert \Im (A)\Vert \} \le w^2(A) \, \le \, w^2( |\Re (A)| +i |\Im (A)|),$$ where $$\mu = \big | \Vert \Re (A)+\Im (A)\Vert -\Vert \Re (A)-\Im (A)\Vert \big |.$$ This improve the existing upper and lower bounds of the numerical radius, namely, $$\begin{aligned} \frac{1}{4} \Vert A^*A+AA^*\Vert \le w^2(A) \le \frac{1}{2} \Vert A^*A+AA^*\Vert . \end{aligned}$$