Norm inequalities involving accretive-dissipative 2 × 2 block matrices

Abstract

Let T11,T12,T21, and T22 be n×n complex matrices, and let T=(T11T12T21T22) be accretive-dissipative. It is shown that if f is an increasing convex function on [0,∞) such that f(0)=0, then⦀f(|T12|2)+f(|T21⁎|2)⦀≤⦀f(|T|2)⦀ for every unitarily invariant norm ⦀⋅⦀. Moreover, if f is an increasing concave function on [0,∞) such that f(0)=0, then⦀f(|T12|2)+f(|T21⁎|2)⦀≤4⦀f(|T|24)⦀ for every unitarily invariant norm ⦀⋅⦀. Among other inequalities for the Schatten p-norms, it is shown that‖T12‖pp+‖T21‖pp≤2p−1‖T11‖pp/2‖T22‖pp/2 for p≥2.