On Some Classes of Weighted Spaces of Weakly Holomorphic Functions
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On Some Classes of Weighted Spaces of Weakly Holomorphic Functions Thai Thuan Quang1 Received: 3 March 2020 / Revised: 6 May 2020 / Accepted: 11 May 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract Let v be a weight on a domain D in a metrizable locally convex space E and F be a complete locally convex space. Denote by Hv (D, F ) the weighted space of F -valued holomorphic functions on D satisfying that v.f is bounded, Av (D) subspace of Hv (D, C) with the unit ball is compact for the open-compact topology. The aim of this paper is to study linearization theorems and approximation properties in several different topologies for weighted spaces Av (D, F ) of functions f ∈ Hv (D, F ) such that u ◦ f ∈ Av (D) for every continuous linear functional u on F . Keywords Locally convex spaces · Infinite dimensional holomorphy · Weighted spaces of holomorphic functions · Linearization · Approximation property Mathematics Subject Classification (2010) 46A03 · 46G20 · 46E50 · 46B28
1 Introduction The most important idea of the linearization problem is that, through it, one identifies a given class of F -valued holomorphic functions defined on an open subset of E with the space of continuous linear mappings from a certain space to F, where E, F are locally convex spaces, i.e., a holomorphic mapping is being identified with a linear operator and so one can pursue the study of the approximation property for spaces of holomorphic mappings by this method. The first linearization result for such spaces was obtained by P. Mazet in the year 1984. The problem has received much attention in the past few years. However, most of the results are directed to spaces of functions between Banach spaces. Six years after the announcement of P. Mazet, J. Mujica obtained a linearization theorem for the space H ∞ (D, F ) of Banach-valued bounded holomorphic mappings defined on an open subset
Thai Thuan Quang
[email protected] 1
Department of Mathematics and Statistics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnam
T.T. Quang
of a Banach space. Using this theorem, he proved several results characterizing the approximation property for H ∞ (D) and its predual space. This study has further been continued by E. Caliskan in more than a recent decade. Weighted spaces of holomorphic functions defined on an open subset of a finite or infinite dimensional Banach space have been studied widely in the literature by several mathematicians. Whereas for the results in the finite dimensional case, we attribute to the contributions of K. D. Bierstedt, J. Bonet, A. Galbis, W. H. Summers, R. G. Meise, Rubel, Shields, etc., the infinite dimensional case was introduced by D. Garcia, M. Maestre, and P. Rueda, and further investigated by M. J. Beltran, E. Jord´a, P. Rueda, M. Gupta, D. Baweja, etc. In this paper, we concentrate our attention on the linearization theorems and approximation property in several different topologies for weighted su
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