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arXiv · 1807.02543

Relatively Uniformly Continuous Semigroups on Vector Lattices

Marko Kandić·Michael Kaplin
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Abstract

In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $\tau_{ru}$. We introduce the notions of strong continuity with respect to $\tau_{ru}$ and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$ and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and $C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real line, the heat semigroup and some Koopman semigroups are relatively uniformly continuous on a variety of spaces.

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Marko Kandić, Michael Kaplin. 2018-07-06. Relatively Uniformly Continuous Semigroups on Vector Lattices. https://arxiv.org/abs/1807.02543

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