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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Semigroups of operators on locally convex spaces

Author: V. A. Babalola
Journal: Trans. Amer. Math. Soc. 199 (1974), 163-179
MSC: Primary 47D05
MathSciNet review: 0383142
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Abstract: Let $ X$ be a complex Hausdorff locally convex topological linear space and $ L(X)$ the family of all continuous linear operators on $ X$. This paper discusses the generation and perturbation theory for $ {C_0}$ semigroups $ \{ S(\xi ):\xi \geqslant 0\} \subset L(X)$ such that for each continuous seminorm $ p$ on $ X$ there exist a positive number $ {\sigma _p}$ and a continuous seminorm $ q$ on $ X$ with $ p(S(\xi )x) \leqslant {e^{^\sigma {p^\xi }}}q(x)$ for all $ \xi \geqslant 0$ and $ x \in X$.

These semigroups are studied by means of a realization of $ X$ as a projective limit of Banach spaces, using certain naturally-defined operators and $ {C_0}$ semigroups on these Banach spaces to connect the present results to the classical Hille-Yosida-Phillips theory.

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Keywords: One-parameter $ {C_0}$ semigroups, infinitesimal generator, semigroup generation, compartmentalized operators, $ {L_\mathcal{A}}(X)$-operators, projective limit operators, generator-perturbation by $ {L_\mathcal{A}}(X)$-operators, Hille-Yosida-Phillips theorem
Article copyright: © Copyright 1974 American Mathematical Society