Wave operators and similarity for some non-selfadjoint operators

The purpose of the present paper is to develop a new method for establishing the similarity of a perturbed operator T(ϰ), formally given by T + ϰV, to the unperturbed operator T. It is basically a “small perturbation” theory, since the parameter ϰ is assumed to be sufficiently small. Otherwise the setting of the problem is rather general; T or V need not be symmetric or bounded, although they are assumed to act in a separable Hilbert space r and ϰ need not be real. The basic assumptions are that the spectrum of T is a subset of the real axis, that V can be written formally as V = B* A, where A is T-smooth and B is T*-smooth (see Definition 1.2), and that A(T − ζ)−1 B* is uniformly bounded for nonreal ζ. Here A and B are (in general unbounded) operators from r to another Hilbert space r′ (r′ = r is permitted). Strictly speaking, we are dealing with a certain extension T(ϰ) of T + ϰB* A, which is uniquely determined by T, A, and B; T(ϰ) = T + ϰB* A is true if A and B are bounded1).