Spaces of Smooth and Generalized Vectors of the Generator of an Analytic Semigroup and Their Applications

Abstract

For a strongly continuous analytic semigroup {e tA }t ≥ 0 of linear operators in a Banach space B, we study some locally convex spaces of smooth and generalized vectors of its generator A, as well as the extensions and restrictions of this semigroup to these spaces. We extend the Lagrange result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on the description of the set of all vectors x∈𝔅 for which the limit $$ {\lim}_{n\to \infty }{\left(I+\frac{tA}{n}\right)}^nx $$ exists and coincides with e tA x. Moreover, we present a brief survey of particular problems whose solutions are required for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator, the representation of solutions of operator-differential equations on an open interval and the investigation of their boundary values, and the existence of solutions of an abstract Cauchy problem in various classes of analytic vector-valued functions.