Operator inequalities related to the Corach–Porta–Recht inequality

Abstract

Using elementary techniques we prove that if A, B are invertible positive operators in B(H ), t ≤ 2 and r ∈ [ 1/2 , 3/2 ], then (2 + t) ||| ArXB2−r + A2−rXBr ≤ 2 ||| A 2X + tAXB + XB2 ||| for any unitarily invariant norm |||.||| and X in the associated ideal J|||.|||. We also characterize the class of operators satisfying || SXS−1 + S −1XS + kX ||≥ (k+2) ||X|| under certain conditions.