An Analogue of the Kato–Rosenblum Theorem for Commuting Tuples of Self-Adjoint Operators

Abstract:

Suppose that A= (A ₁, ..., A _N ) and are tuples of self-adjoint operators on a Hilbert space H such that and for all 1 ≤j, $k≤N. Suppose that there are z ₁, ..., z _N ∋C\R such that belongs to the trace class, . We prove that is unitarily equivalent to . Here, and is the largest invariant subspace on which A can be simultaneously diagonalized modulo the trace class.