Abstract

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces \((X,\tau )\) that are also equipped with a ‘suitable’ auxiliary norm. We introduce the set \(\mathcal {N}\) of \(\tau \)-continuous semi-norms that are bounded by the norm. If \((X,\tau )\) has the property that \(\mathcal {N}\) 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 bi-continuous semigroups and show that they can be applied to semigroups on \((C_b(E),\beta )\) and \((\mathcal {B}(\mathfrak {H}),\beta )\) for a Polish space \(E\) and a Hilbert space \(\mathfrak {H}\) and where \(\beta \) is their respective strict topology.

Highlights

  • The study of Markov processes on complete separable metric spaces (E, d) naturally leads to transition semigroups on Cb(E) that are not strongly continuous with respect to the norm. These semigroups turn out to be strongly continuous with respect to the weaker locally convex strict topology

  • We show that if the so called mixed topology γ = γ (||·||, τ ), introduced by Wiweger [34], has good sequential properties, bi-continuity of a semigroup for τ is equivalent to strong continuity and local equi-continuity for γ

  • We show that the spaces (Cb(E), β) and (B(H), β), where E is a Polish space, H a Hilbert space and where β is their respective strict topology, are strong Mackey and satisfy Condition C. This implies that our results can be applied to Markov transition semigroups on Cb(E) and quantum dynamical semigroups on B(H)

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Summary

Introduction

The study of Markov processes on complete separable metric spaces (E, d) naturally leads to transition semigroups on Cb(E) that are not strongly continuous with respect to the norm Often, these semigroups turn out to be strongly continuous with respect to the weaker locally convex strict topology. Kunze [23,24] studies semigroups of which he assumes that the resolvent can be given in integral form His notions are topological, and he gives a Hille-Yosida theorem for equi-continuous semigroups. 8, we show that the spaces (Cb(E), β) and (B(H), β), where E is a Polish space, H a Hilbert space and where β is their respective strict topology, are strong Mackey and satisfy Condition C This implies that our results can be applied to Markov transition semigroups on Cb(E) and quantum dynamical semigroups on B(H)

Preliminaries
Strong Mackey spaces: connecting strong continuity and local equi-continuity
A suitable structure of bounded sets
Infinitesimal properties of semigroups
Generation results
Relating bi-continuous semigroups to SCLE semigroups
The strict topology
Full Text
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