On the Uniqueness of Wave Operators Associated with Non-Trace Class Perturbations

Voiculescu has previously established the uniqueness of the wave operator for the problem of e⁽⁰⁾-perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal e has the property lim_n→∞ n^-1/2||P_n||e = 0, where {P_n} is any sequence of orthogonal projections with rank(P _n) = n. We prove that the same uniqueness result holds true so long as e is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation).