Disjoint Dynamics on Weighted Orlicz Spaces
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Complex Analysis and Operator Theory
Disjoint Dynamics on Weighted Orlicz Spaces Chung-Chuan Chen1
· Serap Öztop2 · Seyyed Mohammad Tabatabaie3
Received: 3 March 2020 / Accepted: 5 September 2020 © Springer Nature Switzerland AG 2020
Abstract We give some sufficient and necessary conditions for translation operators on the weighted Orlicz spaces to be disjoint topologically transitive and disjoint topologically mixing. In particular, we show that in certain cases, operators are disjoint topologically transitive if, and only if, their direct sum is topologically transitive. Keywords Disjoint topological transitivity · Translation operator · Weighted Orlicz space · Locally compact group Mathematics Subject Classification Primary 47A16; Secondary 46E30 · 54H20
1 Introduction In [11], we characterized topological transitivity and chaos for translation operators on the weighted Orlicz spaces. This note is a continuation of the study in [11] about linear dynamics on the weighted Orlicz spaces. Indeed, we will tackle the deeper notion, namely, disjoint topological transitivity, which was introduced by Bernal-González,
Communicated by H. Turgay Kaptanoglu. This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.
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Chung-Chuan Chen [email protected] Serap Öztop [email protected] Seyyed Mohammad Tabatabaie [email protected]
1
Department of Mathematics Education, National Taichung University of Education, Taichung 403, Taiwan
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Department of Mathematics, Faculty of Science, Istanbul University, Istanbul, Turkey
3
Department of Mathematics, University of Qom, Qom, Iran 0123456789().: V,-vol
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C.-C. Chen et al.
Bès and Peris [5,7] respectively in 2007. The active topic of disjointness was studied intensely by many authors (for instance, see [6,22]). About four decades ago, linear dynamics started to attract a lot of attention. We refer to these two classic books [3,14] on this subject. The mappings on the spaces p (Z), L p (R) and L p (C) provided concrete examples and played important roles in this investigation. We note that these spaces are special cases of the Lebesgue spaces of locally compact groups. Let G be a locally compact groups. The interest of linear dynamics on L p (G) was initiated in [8,10] where the transitive and chaotic weighted translations were studied. Since then, fruitful results on L p (G) appear. Indeed, Chen recently investigated the existence of hypercyclic weighted translations on L p (G) in [13]. Disjoint hypercyclicity of weighted translations on L p (G) was characterized in [9,16]. Let w be a weight on G. From another view, the density of translates in the p weighted Lebesgue space L w (G) was discussed by Abakumov and Kuznetsova in [1]. Inspired by [1], and based on our previous work [11,12], in this paper, we will study the notions of disjoint topological transitivity and disjoint topological mixing on the weighted Orlicz spaces. An operator T o
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