An analogue of the Kato-Rosenblum theorem for commuting tuples of self-adjoint operators

Suppose that A = (A₁, ..., A_N) and A′ = (A′₁, ..., A′_N) are tuples of self-adjoint operators on a Hilbert space H such that [A_j, A_k] = 0 and [A′_j, A′_k] = 0 for all 1 ≤ j, k ≤ N. Suppose that there are z₁, ..., Z_N ∈ C\R such that (A_j - z_j)^-1 - (A′_j - z_j)^-1 belongs to the trace class, 1 ≤ j ≤ N. We prove that A\H_nd(A;C₁) is unitarily equivalent to A′\H_nd(A′;C₁). Here, H = H_d(A;C₁) ⊕ H_nd(A;C₁) and H_d(A;C₁) is the largest invariant subspace on which A can be simultaneously diagonalized modulo the trace class.