Relatively uniformly continuous semigroups of positive operators on vector lattices : doctoral thesis

Kaplin, Michael (author), Kramar Fijavž, Marjeta (mentor), Kandić, Marko (co-mentor)

In this thesis we introduce and study notions of relatively uniform continuity and strong continuity with respect to the relatively uniform topology for semigroups of operators on general vector lattices. 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 ${\rm Lip}({\mathbb R})$, ${\rm UC}({\mathbb R})$, and ${\rm C}_c({\mathbb R})$. We provide examples of relatively uniformly continuous semigroups such as Koopman semigroups and the Ornstein-Uhlenbeck semigroup. We introduce notions of relatively uniformly continuous, differentiable, and integrable functions on ${\mathbb R}_+$ which enable us to study generators of relatively uniformly continuous semigroups. Our main result is a Hille-Yosida type theorem which provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.

• Language: English
• Keywords: vector lattices, relatively uniform convergence, relatively uniform topology, relatively uniform continuity, positive operator semigroups, strongly continuous semigroups, Hille-Yosida theorem
• Work type: Doctoral dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120744
• UDC: 517.9
• COBISS.SI-ID: 32235523
• Publication date in RUL: 25.09.2020

Secondary language

• Language: Slovenian
• Title: Relativno enakomerno zvezne polgrupe pozitivnih operatorjev na vektorskih mrežah

Abstract

V disertaciji uvedemo in obravnavamo pojme relativno enakomerne zveznosti in krepke zveznosti glede na relativno enakomerno topologijo za polgrupe operatorjev na splošnih vektorskih mrežah. Z njihovo pomočjo obravnavamo polgrupe na prostorih, ki niso lokalno konveksni, kot so $L^p({\mathbb R})$ za $0 < p < 1$, in nekompletnih prostorih ${\rm Lip}({\mathbb R})$, ${\rm UC}({\mathbb R})$ in ${\rm C}_c({\mathbb R})$. Predstavimo tudi primere relativno enakomerno zveznih polgrup kot so Koopmanove polgrupe in Ornstein-Uhlenbeckova polgrupa. Predstavimo pojme relativno enakomerno zveznih, odvedljivih in integrabilnih funkcij na ${\mathbb R}_+$. Z njihovo pomočjo obravnavamo generatorje relativno enakomerno zveznih polgrup. Glavni rezultat je izrek tipa Hille-Yosida, ki nudi potrebne in zadostne pogoje, da je operator generator eksponentno urejenostno omejene, relativno enakomerno zvezne in pozitivne polgrupe.

• Keywords: vektorske mreˇze, relativno enakomerna konvergenca, relativno enakomerna topologija, relativno enakomerna zveznost, pozitivne operatorske polgrupe, krepko zvezne polgrupe, izrek Hille-Yosida