Some Properties of Tensorial Perspective for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Abstract

Let H be a Hilbert space. Assume that f:[0,∞)→R is continuous and A, B>0. We define the tensorial perspective for the function f and the pair of operators (A,B) by

P_{f,⊗}(A,B):=(1⊗B)f(A⊗B⁻¹).

In this paper we show among others that, if f is differentiable convex, then

P_{f},_{⊗}(A,B)≥[f(u)-f′(u)u](1⊗B)+f′(u)(A⊗1),

for A, B>0 and u>0. Moreover, if Sp(A)⊂I, Sp(B)⊂J and such that 0<γ≤(t/s)≤Γ for t∈I and s∈J, then

P_{f},_{⊗}(A,B) ≤[f(u)-f′(u)u](1⊗B)+f′(u)(A⊗1)
+[f₋′(Γ)-f₊′(γ)]|A⊗1-u(1⊗B)|

for u∈[γ,Γ].