Cauchy Problems Involving Non-Selfadjoint Operators

We consider the differential equation where for each is f(t) an element of a Hilbert space H and H: dom H → H is a closed, densely defined linear operator with real spectrum. We present a class, the U-scalar operators, which possess a representation of the form on a dense set. Here E(S), s∊R is a family of possibly unbounded commuting projections. U-scalar operators are a generalization of spectral operators of scalar type. If H is U-scalar we show that defines a one parameter group of denselydefined linear operators and that, for a dense set of initial conditions f0, a strong solution of (1) is given by f(t)=B(t)f0. We give necessary and sufficient conditions for an operator to be U-scalar. We prove that U-scalar operators do not possess residual spectrum