Dominated and pointwise ergodic theorems with “weighted” averages for bounded Lamperti representations of amenable groups

Arkady Tempelman, Alexander Shulman

Research output: Contribution to journalArticle

Abstract

Group representations by bounded Lamperti operators in the spaces L α (1≤α<∞) form a wide class of representations, including representations by bounded positive operators and (when α≠2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesàro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in L α (1<α<∞) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989–5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to “weighted” averages of such representations and discuss various conditions on the “weights” under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups R m and Z m .

Original languageEnglish (US)
Pages (from-to)23-58
Number of pages36
JournalJournal of Mathematical Analysis and Applications
Volume474
Issue number1
DOIs
StatePublished - Jun 1 2019

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Amenable Group
Ergodic Theorem
Weighted Average
Bounded Operator
Polynomials
Group Representation
Polynomial Growth
L-space
Locally Compact Group
Positive Operator
Analytical Methods
Isometric
Generalise
Operator
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Group representations by bounded Lamperti operators in the spaces L α (1≤α<∞) form a wide class of representations, including representations by bounded positive operators and (when α≠2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Ces{\`a}ro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in L α (1<α<∞) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989–5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to “weighted” averages of such representations and discuss various conditions on the “weights” under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups R m and Z m .",
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Dominated and pointwise ergodic theorems with “weighted” averages for bounded Lamperti representations of amenable groups. / Tempelman, Arkady; Shulman, Alexander.

In: Journal of Mathematical Analysis and Applications, Vol. 474, No. 1, 01.06.2019, p. 23-58.

Research output: Contribution to journalArticle

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