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

Abstract In this note, we study the problem of evaluating the trace of f ( T ) − f ( R ) , where T and R are contractions on a Hilbert space with trace class difference, i.e. T − R ∈ 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 ) ∈ S 1 . The main result of the note says that there exists a function ξ (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 ) ) = ∫ T f ′ ( ζ ) ξ ( ζ ) d ζ , whenever T and R are contractions with T − R ∈ S 1 , and f is an operator Lipschitz function analytic in D .