Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators A,B, a bounded operator J, and a function f:R→C, we obtain bounds in quasi-normed ideals of compact operators for the difference f(A)J−Jf(B) in terms of the operator AJ−JB. The focus is on functions f that are smooth everywhere except for finitely many points. A typical example is the function f(t)=|t|γ with γ∈(0,1). The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace trf(S) with S being a Wiener–Hopf operator with a discontinuous symbol

Functions of normal operators under perturbations

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the H\"older class \L_\a(\R^2), 0<\a<1, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then \|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a. We obtain a more general result for functions in the space \L_\o(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\} for an arbitrary modulus of continuity \o. We prove that if $f$ belongs to the Besov class B_{\be1}^1(\R^2), then it is operator Lipschitz, i.e., \|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|. We also study properties of $f(N_1)-f(N_2)$ in the case when f\in\L_\a(\R^2) and $N_1-N_2$ belongs to the Schatten-von Neuman class \bS_p.Comment: 32 page

Taylor approximations of operator functions

This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.Comment: 12 page

Trotter-Kato product formulae in Dixmier ideal

It is shown that for a certain class of the Kato functions the Trotter-Kato product formulae converge in Dixmier ideal C 1,$\infty$ in topology, which is defined by the $\times$ 1,$\infty$-norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence. 1 since [24], [14]. Note that a subtle point of this program is the question about the rate of convergence in the corresponding topology. Since the limit of the Trotter-Kato product formula is a strongly continuous semigroup, for the von Neumann-Schatten ideals this topology is the trace-norm $\times$ 1 on the trace-class ideal C 1 (H). In this case the limit is a Gibbs semigroup [25]. For self-adjoint Gibbs semigroups the rate of convergence was estimated for the first time in [7] and [9]. The authors considered the case of the Gibbs-Schr{\"o}dinger semigroups. They scrutinised in these papers a dependence of the rate of convergence for the (exponential) Trotter formula on the smoothness of the potential in the Schr{\"o}dinger generator. The first abstract result in this direction was due to [19]. In this paper a general scheme of lifting the operator-norm rate convergence for the Trotter-Kato product formulae was proposed and advocated for estimation the rate of the trace-nor

On the structure of the essential spectrum of elliptic operators on metric spaces

We give a description of the essential spectrum of a large class of operators on metric measure spaces in terms of their localizations at infinity. These operators are analogues of the elliptic operators on Euclidean spaces and our main result concerns the ideal structure of the $C^*$-algebra generated by them.Comment: Improved presentation, some new results