Higher-Order Trace Formulas for Contractive and Dissipative Operators

Abstract: We establish higher order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator of the contractions and enlarging the set of admissible functions. We also derive higher order trace formulas for maximal dissipative operators under relaxed assumptions and new simplified trace formulas for unitary and resolvent comparable self-adjoint operators. The respective spectral shift measures are absolutely continuous and, in the case of contractions, the set of admissible functions for the $n$th order trace formula on the unit circle includes the Besov class $B^n_{\infty, 1}(\T)$. Both aforementioned properties are new in the mentioned generality.