Higher order spectral shift, II. Unbounded case

TL;DR

This paper develops higher order spectral shift functions for unbounded self-adjoint operators, extending previous bounded operator results and expressing these functions recursively via lower order functions.

Contribution

It introduces a recursive construction of higher order spectral shift functions for unbounded operators, generalizing Krein's and Koplienko's functions.

Findings

Constructed higher order spectral shift functions for unbounded operators.

Extended previous bounded operator results to unbounded case.

Provided recursive formulas relating higher and lower order spectral shift functions.

Abstract

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 operator. In the particular cases of the zero and the first order approximations, the corresponding spectral shift functions have been constructed by M. G. Krein and L. S. Koplienko, respectively. The higher order spectral shift functions obtained in this paper can be expressed recursively via the lower order ones, in particular, Krein's and Koplienko's spectral shift functions. This extends the recent results of Dykema and Skripka for bounded operators.