Further refinements of the Heinz inequality

Abstract

The celebrated Heinz inequality asserts that 2|||A^1/2XB^1/2||| leq|||A^νXB^(1−ν)+ A^(1−ν)XB^ν||| leq |||AX + XB||| for X ∈ B(H ), A,B ∈B(H )_+, every unitarily invariant norm ||| · ||| and ν ∈ [0,1]. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function F(ν) = ||||A^νXB^(1−ν)+ A^(1−ν)XB^ν|||, some integration techniques and various refinements of the Hermite?Hadamard inequality.