Related papers: Higher-Order Trace Formulas for Contractive and Di…
We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being…
We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the…
Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later in 1988, a similar formula was obtained by Neidhardt \cite{NH} in the case of…
In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the differences…
Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to…
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…
We obtain Taylor approximations for functionals $V\mapsto Tr(f(H_0+V))$ defined on the bounded self-adjoint operators, where $H_0$ is a self-adjoint operator with compact resolvent and $f$ is a sufficiently nice scalar function, relaxing…
We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53]…
We consider non-smooth functions of (truncated) Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. Our main results are uniform estimates for trace norms ($d\ge 1$) and quasiclassical asymptotic formulas for traces of the…
We describe the maximal class of functions $f$ on the real line, for which the Lifshitz--Krein trace formula $\operatorname{trace}(f(A)-f(B))=\int_{\Bbb R} f'(s)\boldsymbol{\xi}(s)\,ds$ holds for arbitrary self-adjoint operators $A$ and $B$…
The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of…
Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $\varphi\mapsto\varphi(A,B)$ for functions $\varphi$ in the Besov class…
We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities…
We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function $f$ on ${\Bbb R}^2$, for which the map $(A,B)\mapsto f(A,B)$ is Lipschitz in…
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz…
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They…
We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko on representations for the remainders of the first and second order Taylor-type approximations of operator…
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz…
A first order trace formula is obtained for a regular differential operator perturbed by a finite signed measure multiplication operator.
We construct higher order spectral shift functions, which represent the remainders of Taylor-type approximations for the value of a function at a perturbed self-adjoint operator by derivatives of the function at an initial unbounded…