On the difference of spectral projections

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 $(-\mathrm{\infty },\lambda )$. In the case when $S$ is of rank one, we show that $D(\lambda )$ is unitarily equivalent to a block diagonal operator ${\mathrm{\Gamma }}_{\lambda }\oplus 0$, where ${\mathrm{\Gamma }}_{\lambda }$ 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 ${\mathrm{\ell }}^2({\mathbb{N}}_0)$ for all $ \lambda \in $\mathbb{R}$ $exceptforatmostcountablymany$ \lambda $.