Spaces of smooth and generalized vectors of the generator
of an analytic semigroup and their applications
Authors
V. M. Gorbachuk
M. L. Gorbachuk
Abstract
For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t\geq 0}$ of linear operators in a Banach space $B$ we investigate 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 Lagrange’s 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 description of the set of all vectors
$x \in B$ for which there exists $$\mathrm{l}\mathrm{i}\mathrm{m}_{n\rightarrow \infty }\biggl( I + \frac{tA}n \biggr)^n x$$
and this limit coincides with etAx. Moreover, we present a short
survey of particular problems whose solutions are necessary 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 to operator-differential equations on an open interval and the analysis of their boundary values, and the existence
of solutions to an abstract Cauchy problem in various classes of analytic vector-valued functions.
