Golden-Thompson type inequalities related to a geometric mean via specht's ratio

Abstract

We prove a Golden-Thompson type inequality via Specht’s ratio: Let H and K be selfadjoint operators on a Hilbert space H satisfying MI H, K mI for some scalar M > m . Then Mh(1) ( (1 − λ)etH + λetK ) 1 t e(1−λ )H+λK Mh(1)−1Mh(t)− 1 t ( (1 − λ)etH + λetK ) 1 t holds for all t > 0 and 0 λ 1 , where h = eM−m and (generalized) Specht’s ratio Mh(t) is defined for h > 0 as Mh(t) = (ht − 1)h t ht−1 e log ht (h = 1) and M1(1) = 1. Mathematics subject classification (2000): 47A30, 47A63.