Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations
Arup Chattopadhyay (Indian Institute of Technology Guwahati)
Teun D.H. van Nuland (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Chandan Pradhan (Indian Institute of Science)
More Info
expand_more
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.
Abstract
Given H self-adjoint, V symmetric and relatively H-bounded, and f:R→C satisfying mild conditions, we show that the Gateaux derivativedndtnf(H+tV)|t=0 exists in the operator norm topology, for every natural n, and establish perturbation formulas for multiple operator integrals under relatively bounded perturbations. If the H-bound of V is less than 1, we obtain sufficient conditions on f which ensure that the Taylor expansionf(H+V)=∑n=0∞1n!dndtnf(H+tV)|t=0 exists and converges absolutely in operator norm. Assuming that V(H−i)−p∈Ss/p for p=1,…,s for some s∈N[jls-end-space/], we show that the Krein–Koplienko spectral shift functions ηk,H,V[jls-end-space/], satisfying(Formula presented) exist for every k=1,2,3,…[jls-end-space/], independently of s. The latter result (which is significantly stronger than [27]) is new also in the case that V is bounded. The proof is based on [34], combined with a generalisation of the multiple operator integral compatible with [17]. We discuss applications of our results to quantum physics and noncommutative geometry.
help_outline