Title: Higher order spectral shift, II. Unbounded case.

Authors: Anna Skripka

Issue: Volume 59 (2010), Issue 2, 691-706

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 [M.G. Krein, \textit{On the trace formula in perturbation theory}, Mat. Sbornik N.S. \textbf{33(75)} (1953), 597-626] and L. S. Koplienko [L.S. Koplienko, \textit{The trace formula for perturbations of nonnuclear type}, Sibirsk. Mat. Zh. \textbf{25} (1984), 62-71], 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 [K. Dykema and A. Skripka \textit{Higher order spectral shift}, J. Funct. Anal. \textbf{257} (2009), 1092-1132] for bounded operators.