On the difference of spectral projections

TL;DR

This paper investigates the spectral projection differences caused by compact perturbations of self-adjoint operators, revealing their unitary equivalence to Hankel operators for almost all spectral parameters.

Contribution

It establishes that the difference of spectral projections under compact perturbations is unitarily equivalent to Hankel operators, extending known results from rank-one to general compact operators.

Findings

Spectral projection differences are unitarily equivalent to Hankel operators.

Results hold for all but countably many spectral parameters.

Generalizes from rank-one to arbitrary compact perturbations.

Abstract

For a semibounded self-adjoint operator $ T $ and a compact self-adjoint operator $ S $ acting on a complex separable Hilbert space of infinite dimension, we study the difference $ D(\lambda) := E_{(-\infty, \lambda)}(T+S) - E_{(-\infty, \lambda)}(T), \, \lambda \in \mathbb{R} $, of the spectral projections associated with the open interval $ (-\infty, \lambda) $. In the case when $ S $ is of rank one, we show that $ D(\lambda) $ is unitarily equivalent to a block diagonal operator $ \Gamma_{\lambda} \oplus 0 $, where $ \Gamma_{\lambda} $ is a bounded self-adjoint Hankel operator, for all $ \lambda \in \mathbb{R} $ except for at most countably many $ \lambda $. If, more generally, $ S $ is compact, then we obtain that $ D(\lambda) $ is unitarily equivalent to an essentially Hankel operator (in the sense of Mart\'{\i}nez-Avenda\~no) on $ \ell^{2}(\mathbb{N}_{0}) $ for all $ \lambda \in \mathbb{R} $ except for at most countably many $ \lambda $.