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.