Let L(H) be the algebra of bounded operators on a complex separable Hilbert space H. Let N be a unitarily invariant norm defined on a norm ideal J ⊆ L(H). Given two positive invertible operators P,Q ∊ L(H) and k ∊ (−2, 2], we show that N (PTQ−1 + P−1TQ + kT) ≥ (2 + k)N(T), T ∊ J. 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 γ : L(H) → L(H) given by γ(T) = PTQ−1 +P−1TQ+kT is invertible, and we estimate the induced norm of γ−1 acting on the norm ideal J. We compute sharp constants for the involved inequalities in several particular cases.Universidad del ComahueFacultad de Ciencias Exactas