  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 $.
