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 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 B for which there exists lim_n→ ∞ ( I + tAn )^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.