Weighted Composition Semigroups on Spaces of Continuous Functions and Their Subspaces

Pub Date : 2024-06-11 DOI:10.1007/s11785-024-01559-5
Karsten Kruse
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Abstract

This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces \(\mathcal {F}(\Omega )\) of continuous functions on a Hausdorff space \(\Omega \) such that the norm-topology is stronger than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces, the space of bounded Dirichlet series and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Gutiérrez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces \(\mathcal {F}(\mathbb {D})\) of holomorphic functions on the open unit disc \(\mathbb {D}\) such that \(H^{\infty }\subset \mathcal {F}(\mathbb {D})\subset \mathcal {B}_{1}\) where \(H^{\infty }\) is the Hardy space of bounded holomorphic functions on \(\mathbb {D}\) and \(\mathcal {B}_{1}\) the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity.

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连续函数空间及其子空间上的加权合成半群
本文致力于连续函数空间及其子空间上的加权组成半群。我们考虑在连续函数的巴拿赫空间(Banach space \(\mathcal {F}(\Omega )\) on a Hausdorff space \(\Omega \))上由半流和半循环诱导的半群,这些半群的规范拓扑强于紧凑开式拓扑,比如哈代空间(Hardy spaces)、加权伯格曼空间(the weighted Bergman spaces)、狄里克特空间(the Dirichlet space)、布洛赫类型空间(the Bloch type spaces)、有界狄里克特数列空间(the space of bounded Dirichlet series)和连续函数或全形函数的加权空间(the weighted spaces of continuous or holomorphic functions)。加利亚多-古铁雷斯、西斯卡基斯和雅库布证明了这一点、西斯卡基斯和雅库博维奇证明,不存在非难规范-上的强连续加权组成半群,这样(H^{infty }\subset\其中 \(H^{\infty }\) 是 \(\mathbb {D}\) 上有界全形函数的哈代空间,而 \(\mathcal {B}_{1}\) 是布洛赫空间。然而,我们证明了在这些空间上存在非难加权组成半群,它们在规范拓扑和紧凑开式拓扑之间的混合拓扑中是强连续的。我们在连续函数的巴拿赫空间的一般环境中研究这种加权组成半群,并推导出在混合拓扑中强连续性所涉及的空间、半流和半环的必要和充分条件,以及规范强连续性的副产品。此外,我们还给出了它们的生成器和它们的强规范连续性空间的几个特征。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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