A Hille-Yosida theorem for Bi-continuous semigroups

Kühnemund, Franziska

doi:10.1007/s00233-002-5000-3

Cited by 71 publications

(155 citation statements)

References 18 publications

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“…The proof of the Hille-Yosida theorem stated here, however, gives explicit control on the semigroup rescaled by e −ωt via the construction in Lemma 6.2 and gives a result as strong as the equivalence of (a) and (b) of Theorem 16 in [22]. First note that we can always assume that ω = 0 by a suitable rescaling.…”

Section: Corollary 65 Suppose That (X τ ||·||) Satisfies Conditionmentioning

confidence: 90%

“…We have now developed enough machinery to prove a Hille-Yosida type theorem which resembles the equivalence between (a) and (b) of Theorem 16 in [22]. A, D(A)) is closed, densely defined and there exists ω ∈ R and M ≥ 1 such that for every λ > ω one has λ ∈ ρ(A) and for every semi-norm p ∈ N and λ 0 > ω there exists a semi-norm q ∈ N such that for all x ∈ X one has A, D(A)) is closed, densely defined and there exists ω ∈ R and M ≥ 1 such that for every λ ∈ C satisfying Re λ > ω, one has λ ∈ ρ(A) and for every semi-norm p ∈ N and λ 0 > ω there exists a semi-norm q ∈ N such that for all x ∈ X and n ∈ N sup…”

Section: Lemma 63 Let (X τ ||·||) Satisfy Condition C Letmentioning

confidence: 99%

“…Bi-continuous semigroups were introduced by Kühnemund [22] to study semigroups on Banach spaces that are strongly continuous with respect to a weaker locally convex topology τ and where τ has good sequential properties on norm bounded sets. We will consider the mixed topology γ :=γ (||·|| , τ ), introduced by Wiweger [34], also see [9], which is the strongest locally convex topology that coincides with τ on norm bounded sets.…”

Section: Relating Bi-continuous Semigroups To Scle Semigroupsmentioning

confidence: 99%

“…Essentially, these spaces are also equipped with a norm ||·|| such (X, ||·||) is Banach and such that the dual (X, τ ) is norming for (X, ||·||). Bi-continuous semigroups have been studied in [3,14,22], in which the Hille-Yosida, Trotter Approximation theorem and perturbation results have been shown. Bi-continuity has the drawback, however, that it is a non-topological notion.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Corollary 65 Suppose That (X τ ||·||) Satisfies Conditionmentioning

confidence: 90%

Section: Lemma 63 Let (X τ ||·||) Satisfy Condition C Letmentioning

confidence: 99%

Section: Relating Bi-continuous Semigroups To Scle Semigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kraaij

2015

Semigroup Forum

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“…They only consider the situation where E is a Hilbert space, in which case Itô calculus may be applied. Using analytic methods, the Banach space case was studied in [10], [12], [13], [9]. We will need the following result from [13], which is an easy consequence of Proposition 2.1 and (3.1).…”

Section: The Lie-trotter Product Formulamentioning

confidence: 99%

A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

Kühnemund

Neerven²

2004

Journal of Evolution Equations

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A Desch–Schappacher perturbation theorem for bi‐continuous semigroups

Budde

Farkas

2020

Mathematische Nachrichten

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We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra ℒ( ) of bounded, linear operators on a Banach space , and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem. K E Y W O R D Sbi-continuous semigroups, Desch-Schappacher type perturbation, extrapolation spaces, implemented semigroups M S C ( 2 0 1 0 ) 34G10, 46A70, 47A55, 47D03 INTRODUCTIONAs suggested by Greiner in [17] abstract perturbation theory of one-parameter semigroups provides good means to change the domain of a semigroup generator. For this an enlargement of the underlying Banach space may be necessary and extrapolation spaces become important. One of the well-known results in this direction goes back to the papers of Desch and Schappacher, see [7] and [8]. Another prominent example of such general perturbation techniques is due to Staffans and Weiss, [30,31], and an elegant abstract operator theoretic/algebraic approach has been developed by Adler, Bombieri and Engel in [1]. A general theory of unbounded domain perturbations is given by Hadd, Manzo and Rhandi [18]. A more recent paper by Bátkai, Jacob, Voigt and Wintermayr [4] extends the notion of positivity to extrapolation spaces, and studies positive perturbations for positive semigroups on AM-spaces. Hence, the study of abstract Desch-Schappacher type perturbations is a lively research field, to which we contribute with the present article. The reason for such an active interest in this area is that the range of application is vast. We mention here only a selection from the most recent ones: boundary perturbations by Nickel [29], boundary feedback by Casarino, Engel, Nagel and Nickel [6], boundary control by Engel, Kramar Fijavž, Klöss, Nagel and Sikolya [11] and Engel and Kramar Fijavž [10], port-Hamiltonian systems by Baroun and Jacob [3], control theory by Jacob, Nabiullin, Partington and Schwenninger [19,20] and Jacob, Schwenninger and Zwart [21] and vertex control in networks by Engel and Kramar Fijavž [9,12].All the previously mentioned abstract perturbation results were developed for strongly continuous semigroups of linear operators on Banach spaces, 0 -semigroups for short. This is, for certain applications, e.g., for the theory of Markov transition semigroups (see [27], for a more general theory we refer to [26]), far too restrictive. For this situation the Banach space of

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A Hille-Yosida theorem for Bi-continuous semigroups

Cited by 71 publications

References 18 publications

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

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

A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

A Desch–Schappacher perturbation theorem for bi‐continuous semigroups

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