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 $$ \Gamma _{\lambda } + \Lambda _{\lambda } $$ for all $$ \lambda \in \mathbb {R} $$ except for at most countably many $$ \lambda $$ , where $$ \Gamma _{\lambda } $$ is a bounded self-adjoint Hankel operator and $$ \Lambda _{\lambda } $$ is a compact self-adjoint operator.