Uniform ergodic theorems for semigroup representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929462913794048 |
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| author | Glück, Jochen Hermle, Patrick Kreidler, Henrik |
| author_facet | Glück, Jochen Hermle, Patrick Kreidler, Henrik |
| contents | We consider a bounded representation $T$ of a commutative semigroup $S$ on a Banach space and analyse the relation between three concepts: (i) properties of the unitary spectrum of $T$, which is defined in terms of semigroup characters on $S$; (ii) uniform mean ergodic properties of $T$; and (iii) quasi-compactness of $T$.
We use our results to generalize the celebrated Niiro-Sawashima theorem to semigroup representations and, as a consequence, obtain the following: if a positive and bounded semigroup representation on a Banach lattice is uniformly mean ergodic and has finite-dimensional fixed space, then it is quasi-compact. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_08961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform ergodic theorems for semigroup representations Glück, Jochen Hermle, Patrick Kreidler, Henrik Functional Analysis Spectral Theory 47D03, 47D06, 47B65, 46B42 We consider a bounded representation $T$ of a commutative semigroup $S$ on a Banach space and analyse the relation between three concepts: (i) properties of the unitary spectrum of $T$, which is defined in terms of semigroup characters on $S$; (ii) uniform mean ergodic properties of $T$; and (iii) quasi-compactness of $T$. We use our results to generalize the celebrated Niiro-Sawashima theorem to semigroup representations and, as a consequence, obtain the following: if a positive and bounded semigroup representation on a Banach lattice is uniformly mean ergodic and has finite-dimensional fixed space, then it is quasi-compact. |
| title | Uniform ergodic theorems for semigroup representations |
| topic | Functional Analysis Spectral Theory 47D03, 47D06, 47B65, 46B42 |
| url | https://arxiv.org/abs/2408.08961 |