Scattering matrix and spectral shift function of the nuclear dissipative scattering theory

The paper is devoted to the problem of existence of the spectral shift function for a pair of contractions which differ by a nuclear operator, it is well-known that the usual spectral shift function for a pair of unitary operators does not allow an extension to this case in general and, moreover, the shift function cannot be obtained as a summable one. This fact yields for instance that the famous trace formula cannot be verified in this situation. In the paper the notion of the spectral shift function is generalized in an appropriate manner such that the shift is summable and satisfies a modified trace formula. Moreover, a modification of the Birman-Krein formula is established. All results are transformed to a pair of maximal dissipative operators whose resolvent difference belongs to the trace class