2023/12/14 by Tirthankar Bhattacharyya, Bhattacharyya, Tirthankar, Arup Chattopadhyay +5
Mathematics · #47A20 #47A55 #47A56 #47B10 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2312.08706
openalex publication_date 2023/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this note, we provide an elementary proof for the expression of f(U)-f(V) in the form of a double operator integral for every Lipschitz function f on the unit circle \cir and for a pair of unitary operators (U,V) with U-V\inS2(\hilh) (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten 2-Lipschitz estimate ‖f(U)-f(V)‖2≤ ‖f‖\lip(\cir)‖U-V‖2 for all Lipschitz functions f:\cir→\C. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each p∈(1,∞) and for every pair of contractions (T0,T1) with ‖T0‖<1, there exists a constant df, p,T0>0 such that ‖f(T1)-f(T0)‖p≤ df,p, T0‖T1-T0‖p for all Lipschitz functions on \cir. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.