I am going to speak about recent joint results with A. B. Aleksandrov. It is well known that a Lipschitz function does not have to be operator Lipschitz. In other words, the inequality |f(x)-f(y)| = const |x-y| does not imply that ||f(A)-f(B)|| = const ||A-B|| for self-adjoint operators A and B. It turned out that the situation dramatically changes if we consider functions in Hoelder--Zygmund classes. We prove that if 0 = a = 1 and f is in the Hoelder class ?a(R), then for arbitrary self-adjoint operators A and B with bounded A-B, the operator f(A)-f(B) is bounded and ||f(A)- f(B)|| = const ||A-B||a. We prove a similar result for functions f of the Zygmund class ?1(R): ||f(A+K)-2f(A)+f(A-K)|| = const||K||, where A and K are self-adjoint operators. Similar results also hold for all Hoelder-Zygmund classes ?a(R), a > 0. We also study properties of the operators f(A)-f(B) for f in ?a(R) and self-adjoint operators A and B such that A-B belongs to the Schatten--von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.


Prof. Vladimir Peller

Research Area

Pure Maths Seminar


Michigan State University


Tue, 03/03/2009 - 12:00pm