Noncommutative Perspectives of Operator Monotone Functions in Hilbert Spaces

Abstract

Assume that the function f:[0,∞)→R is operator monotone in [0,∞) and has the representation

f(t)=f(0)+bt+∫₀^{∞}((tλ)/(t+λ))dw(λ),

where b≥0 and w is a positive measure on (0,∞). In this paper we obtained among others that

P_{f}(B,P)-P_{f}(A,P)
=b(B-A)+∫₀^{∞}λ²[∫₀¹P((1-t)A+tB+λP)â»Â¹(B-A)┊
┊×((1-t)A+tB+λP)â»Â¹Pdt]dw(λ)

for all A, B, P>0. Applications for weighted operator geometric mean and relative operator entropy are also provided.