Equivalence up to a Rank One Perturbation

This note is devoted to the spectral analysis of rank one perturbations of unitary and self-adjoint operators. We study the following question: given two cyclic (i.e., having simple spectrum) operators A and B, when is A equivalent to B up to a rank one perturbation? More precisely, when does there exist a unitary operator U such that rank (UAU∗−B) = 1? As usual, we are looking for an answer in terms of the spectra of A and B. An analogous question for compact perturbations is answered by the Weyl-von Neumann Theorem [K]. It says that A is equivalent to B+K for some compact K iff the essential spectra σess(A) and σess(B) coincide. A necessary and sufficient condition for A and B to be equivalent up to a trace class operator, which was found by Carey and Pincus [CP], is more delicate and involves additional spectral invariants. In addition to the essential spectra, the isolated eigenvalues of A and B must now obey certain rules. In this paper we make the last step down the ladder and study the case when A and B are equivalent up to a rank one perturbation. A general necessary and sufficient condition in these settings seems out of reach: It is impossible to formulate in any reasonable terms. However, it is still possible to achieve a good assessment of the situation by fully describing the most important particular case. It is quite well understood how isolated eigenvalues of an operator behave under rank one perturbations. On the other hand, by the Weyl-von Neumann Theorem, if A is equivalent to B up to a rank one perturbation, then σess(A) = σess(B). In particular, rank one perturbations do not affect absolutely continuous spectrum. Hence, it seems reasonable to restrict our attention to the case when A and B have singular spectrum and