2017/05/12 by M. M. Malamud, Hagen Neidhardt, Malamud, Mark +3
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1705.04782
openalex publication_date 2017/05/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this note we study the problem of evaluating the trace of f(T)-f(R),\nwhere T and R are contractions on Hilbert space with trace class\ndifference, i.e., T-R\∈ boldsymbolS1 and f is a function analytic in\nthe unit disk Bbb D. It is well known that if f is an operator Lipschitz\nfunction analytic in Bbb D, then f(T)-f(R)\∈ boldsymbolS1. The main\nresult of the note says that there exists a function boldsymbol\ξ (a\nspectral shift function) on the unit circle Bbb T of class L1( Bbb T)\nsuch that the following trace formula holds:\n\trace(f(T)-f(R))=\∫ Bbb T\nf'(\ζ) boldsymbol\ξ(\ζ) ,d\ζ, whenever T and R are\ncontractions with T-R\∈ boldsymbolS1 and f is an operator Lipschitz\nfunction analytic in Bbb D.\n