Euclidean Operator Radius Inequalities of a Pair of Bounded Linear Operators and Their Applications

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, 1/4 parallel to A* A + AA* parallel to + mu/2 max{parallel to R(A)parallel to, parallel to F(A)parallel to} <= w(2) (A) <= w(2)(vertical bar R(A)vertical bar + i vertical bar F(A)vertical bar), where mu = vertical bar parallel to R (A)+ F(A)parallel to - parallel to R(A) - F(A)parallel to vertical bar. This improve the existing upper and lower bounds of the numerical radius, namely, 1/4 parallel to A* A + AA* parallel to + w(2) (A) <= 1/2 parallel to A* A + AA*parallel to.