Local conditions for the existence of the spectral shift function

L. S. Koplienko

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 102-117

Résumé

Let $U_0$, $U_1$ be unitary operators in a Hilbert space. If the operator $U_1-U_0$ is nuclear, then (as M. G. Krein established) there exists a function $\eta$ on the unit circle $\mathbf T$, $\eta=\eta(U_1,U_0)$, $\eta\in L_1(\mathbf T)$ satisfying the equality \begin{gather} tr(\varphi(U_1)-\varphi(U_0))=\int_{\mathbf T}\eta(\zeta)\varphi'(\zeta)d\zeta \end{gather} for all functions $\varphi$ with derivative $\varphi'$ from the Wiener class. M. Sh. Rirman and M. G. Krein proved that the function $\varphi'$ is connected with the scattering matrix $S$ for the pair $U_0$, $U_1$ by \begin{gather} \det S(\zeta)=\exp(-2\pi i\eta(\zeta)), \tag{2} \end{gather} In this paper (1) and (2) are proved under more general (local) conditions on the pair $U_0$, $U_1$. Under these conditions we investigate some properties of the function n and describe the class of functions $\eta$, which are admissible in (1). Applications to differential operators are given.