Some operator inequalities for unitary invariant norms

Abstract

Let L(H) be the algebra of bounded operators on a complex separable Hilbert space H. Let N be a unitary invariant norm defined on a norm ideal I ⊆ L(H)$. Given two positive invertible operators P, Q ∈ L(H) and k ∈ (-2,2], we show that N(PTQ^{-1} +P^{-1}TQ + kT) (2+k) N(T), T ∈ I. This extends Zhang's inequality for matrices. We prove that this inequality is equivalent to two particular cases of itself, namely P=Q and Q=P^{-1}. We also characterize those numbers k such that the map ϒ : L(H)→ L(H) given by ϒ (T) = PTQ^{-1} +P^{-1}TQ + kT is invertible, and we estimate the induced norm of ϒ^{-1} acting on the norm ideal I. We compute sharp constants for the involved inequalities in several particular cases.