Spectral shift for relative Schatten class perturbations

Abstract

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI{)}^{-1}$ belongs to a Schatten–von Neumann ideal ${\mathcal{S}}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $V\in {\mathcal{S}}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.