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The spectral shift function, the characteristic function of a contraction, and a generalized integral
A. V. Rybkin
Abstract:
Let
$T$ be a contraction that is a trace class perturbation of a unitary operator
$V$, and let
$\{\lambda_k\}$ be the discrete spectrum of
$T$. For a sufficiently large class of functions
$\Phi$ the trace formula
$$
\operatorname{tr}\{\Phi(T)-\Phi (V)\}=\sum_k\{\Phi(\lambda_k)-\Phi(\lambda_k/|\lambda_k|)\}+(B)\int_0^{2\pi}\Phi'(e^{i\varphi})\,d\Omega(\varphi),
$$
holds. This formula is a direct analogue of the well-known M. G. Krein trace formula for unitary operators. It is natural to call the function
$\Omega$ the spectral shift distribution. Generally speaking, it is not of bounded variation; however, the integral in the trace formula exists in the wider
$B$-sense. In the present paper an explicit representation is obtained
for
$\Omega$ in terms of the characteristic function
$\Theta(\lambda)$ of
the contraction
$T$, and also a relation between a certain derivative
$\Omega'$ and the scattering matrix
$S(\varphi)$ of the pair
$(T,V)$:
$$
\det S(\varphi)=\exp\{-2\pi i\overline{\Omega'(\varphi)}\,\} \quad \textrm{a.e.\ with respect to Lebesgue measure}
$$
is established. A necessary and sufficient condition that
$\Omega$ have bounded variation is obtained. In particular, the necessary and sufficient condition requires that the singular spectrum of the contraction
$T$ be empty. The main results are complete.
UDC:
517
MSC: Primary
47A45,
47A60; Secondary
47A40 Received: 03.09.1993
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