Lipschitz functions of self-adjoint operators in perturbation theory (Q1096869)

Let H be a Hilbert space, and A a fixed bounded self-adjoint operator on H. In this note it is shown that the inequality \(\| f(A)-f(B)\| \leq c_ f\| A-B\|\) holds for each differentiable function f and each bounded self-adjoint operator B with a constant \(c_ f\) depending only on f if and only if the spectrum \(\sigma\) (A) of A is a finite set. If this condition is satisfied, then one can take \(c_ f=8(\log_ 2m+2)\) 2[f], where m is the cardinality of \(\sigma\) (A), and [f] is the Lipschitz constant of f.