The spectral shift function, the~characteristic function of a~contraction, and a~generalized integral

A. V. Rybkin

Sbornik. Mathematics, Tome 83 (1995) no. 1, pp. 237-281

Résumé

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.