On the theory of semigroups of operators on locally convex spaces

“…For example, in the above example, formally writing the resolvent corresponding to the semigroup in its integral form, yields a function which is not in C ∞ c (R). One can work around this problem, see for example [10,19,26] which were already mentioned in the introduction.…”

Section: Proposition 34 a Semigroup {T (T)} T≥0 Of Continuous Operatmentioning

confidence: 99%

“…Notably, the integral representation of the resolvent is not necessarily available. To solve this problem, Kōmura [26],Ōuchi [10] and Dembart [19] have studied various generalised resolvents. More recently, Albanese and Kühnemund [2] also study asymptotic pseudo resolvents and give a Trotter-Kato approximation result and the Lie-Trotter product formula.…”

Section: Introductionmentioning

confidence: 99%

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

Kraaij

2015

Semigroup Forum

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We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X, τ ) that are also equipped with a 'suitable' auxiliary norm. We introduce the set N of τ -continuous semi-norms that are bounded by the norm. If (X, τ ) has the property that N is closed under countable convex combinations, then a number of Banach space results can be generalised in a straightforward way. Importantly, we extend the Hille-Yosida theorem. We relate our results to those on bicontinuous semigroups and show that they can be applied to semigroups on (C b (E), β) and (B(H), β) for a Polish space E and a Hilbert space H and where β is their respective strict topology.

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“…Another interesting reference in this direction is [Dem74]. The conditions on asymptotic resolvents are realized to be some Palais-Wiener conditions for distributional Laplace transforms to stem from a smooth semigroup.…”

Section: Remarks To Existing Literaturementioning

confidence: 99%

Hille-yosida theory in convenient analysis

Teichmann¹

2002

Rev. Mat. Complut.

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A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.

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On the theory of semigroups of operators on locally convex spaces

Cited by 35 publications

References 11 publications

Desch‐Schappacher perturbation of one‐parameter semigroups on locally convex spaces

Desch‐Schappacher perturbation of one‐parameter semigroups on locally convex spaces

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

Hille-yosida theory in convenient analysis

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