Order boundedness and order continuity properties of positive operator semigroups
Autor: | Glück, Jochen, Kaplin, Michael |
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Rok vydání: | 2022 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandi\'c, Kramar-Fijav\v{z}, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting - where both ruc semigroups and $C_0$-semigroups are well-defined concepts - and compare both notions. We show that the ruc semigroups are precisely those positive $C_0$-semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive $C_0$-semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence. Comment: 13 pages. This is version 2. Minor changes compared to version 1 |
Databáze: | arXiv |
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