Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications
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 Lag...
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| Date: | 2017 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1711 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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