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Operators on Hilbert Space

Source: Miklos Schweitzer 2015,P10

April 8, 2016
Functional Analysiscollege contestsspectral analysis

Problem Statement

Let f:R→Rf:\mathbb{R}\to \mathbb{R} be a continuously differentiable,strictly convex function.Let HH be a Hilbert space and A,BA,B be bounded,self adjoint linear operators on HH.Prove that,if f(A)−f(B)=f′(B)(A−B)f(A)-f(B)=f'(B)(A-B) then A=BA=B.
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