Matrix Measures and Finite Rank Perturbations of Self-adjoint Operators

TL;DR

This paper extends the Aronszajn--Donoghue theorem to finite rank perturbations of self-adjoint operators using matrix-valued measures, introducing vector mutual singularity and generalizing spectral averaging.

Contribution

It introduces the concept of vector mutual singularity for matrix-valued measures and generalizes key spectral theorems to finite rank perturbations.

Findings

Mutual singularity holds for the singular parts of spectral measures under finite rank perturbations.

The notion of vector mutual singularity is effective for matrix-valued measures.

Spectral representation involves the matrix Muckenhoupt A2 condition.

Abstract

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new perturbation theoretic results. Our main result is a generalization of the Aronszajn--Donoghue theorem about the mutual singularity of the singular parts of the spectrum for rank one perturbations to the case of finite rank perturbations. Simple direct sum type examples indicate that an exact generalization is not possible. However, in this paper we introduce the notion of \emph{vector mutual singularity} for the matrix-valued measures and show that if we use this notion, the mutual singularity still holds for the finite rank perturbations. As for the scalar spectral measures and the classical mutual singularity, we show that the singular parts are mutually singular for almost all perturbations. One of the ways to prove that is to use a generalization of the Aleksandrov's spectral averaging to the matrix-valued measures, which is also one of the main results of this paper. Finally, the spectral representation of the perturbed operator is obtained. The matrix Muckenhoupt $A_2$ condition appears naturally there, and it plays an important role in establishing the vector mutual singularity of the spectral measures.