On the difference of spectral projections

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 $.