Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications - Gorbachuk, V. M.; Gorbachuk, M. L. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t≥ 0}$ of linear operators in a Banach space $B$ we investigate somelocally 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 formof 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 $$lim_{n→ ∞ }\biggl( I + \frac{tA}n \biggr)^n x$$ and this limit coincides with etAx. Moreover, we present a shortsurvey of particular problems whose solutions are necessary for the introduction of the above-mentioned spaces, namely, thedescription of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator; the representation ofsolutions to operator-differential equations on an open interval and the analysis of their boundary values, and the existenceof solutions to an abstract Cauchy problem in various classes of analytic vector-valued functions.