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\'ınez-Avendaño) on ${\ell }^2({\mathbb{N}}_0)$ for all $\lambda \in \mathbb{R}$ except for at most countably many $\lambda$.

23 pages, 1 figure. This paper generalizes the results from arXiv:1406.6516v1