Perturbation of the continuous spectrum and unitary equivalence

Abstract

l Introduction. Suppose that A and B are self-ad joint operators in a Hubert space H such that B-A=P is a completely continuous operator. We shall concern ourselves with the problem of finding conditions sufficient to guarantee that B is unitarily equivalent to A. Clearly a necessary condition is that the spectrum of A (considered as a point set on the real line) is equal to the spectrum of B. However this condition is not sufficient; von Neumann