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 $(-\infty, \lambda)$. 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 $\lambda$.