A trace formula for functions of contractions and analytic operator Lipschitz functions

作者：Malamuda Mark; Neidhardt Hagen; Peller Vladimir^*

来源：Comptes Rendus Mathematique, 2017, 355(7): 806-811.

DOI：10.1016/j.crma.2017.06.003

In this note, we study the problem of evaluating the trace of f(T) - f(R), where T and Rare contractions on a Hilbert space with trace class difference, i.e. T - R is an element of S-1, and f is a function analytic in the unit disk D. It is well known that if f is an operator Lipschitz function analytic in D, then f(T) - f(R) is an element of S-1, The main result of the note says that there exists a function xi (a spectral shift function) on the unit circle T of class L-1(T) such that the following trace formula holds: trace(f(T) - f(R)) = integral(T)f '(zeta) xi (zeta) d zeta,whenever T and Rare contractions with T - R is an element of S-1, and fis an operator Lipschitz function analytic in D.

• 出版日期2017-7