TY - JOUR AU - V. M. Gorbachuk AU - M. L. Gorbachuk PY - 2017/04/25 Y2 - 2024/03/29 TI - Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 69 IS - 4 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/1711 AB - For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t\geq 0}$ 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 $$\mathrm{l}\mathrm{i}\mathrm{m}_{n\rightarrow \infty }\biggl( I + \frac{tA}n \biggr)^n x$$ and this limit coincides with etAx. Moreover, we present a 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. ER -