Perturbations and operator trace functions

2010/12/15 by Walter D. van Suijlekom, van Suijlekom, Walter D. · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #hep-th #math-ph #math.FA #math.MP

paper · pdf · doi:10.48550/arxiv.1012.3306

13 pages; To appear in J. Funct. Anal

arxiv created 2010/12/15 · openalex publication_date 2010/12/15 · arxiv updated 2010/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spectral functional tr f(D+A) for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives.

Cited by

• Quadratic Forms, Real Zeros and Echoes of the Spectral Action
• Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets

Related

• Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations
• Asymptotic expansions for trace functionals
• Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. Relatively operator Lipschitz functions
• Functions of compact operators under trace class perturbations
• The behavior of functions of operators under perturbations