The Compact Approximation Property for Weighted Spaces of Holomorphic Mappings
In this paper, we examine the compact approximation property for the weighted spaces of holomorphic functions. We show that a Banach space E has the compact approximation property if and only if the predual \(\mathcal {G}_v(U)\) of the space \(H_v(U)\) co
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Abstract In this paper, we examine the compact approximation property for the weighted spaces of holomorphic functions. We show that a Banach space E has the compact approximation property if and only if the predual Gv (U ) of the space Hv (U ) consisting of all holomorphic mappings f : U → C (complex plane) with sup v(x) f (x) < ∞ has the compact approximation property, where v is a radial x∈U
weight defined on a balanced open subset U of E such that Hv (U ) contains all the polynomials. We have also studied the compact approximation property for the weighted (LB)-space V H (E) of holomorphic mappings and its predual V G(E) for a countable decreasing family V of radial rapidly decreasing weights on E. Keywords Weighted spaces of holomorphic mappings · Approximation property · Compact approximation property 2010 AMS Math. Subject Classification Primary 46G20 · 46E50 · Secondary 46B28
1 Introduction The approximation property plays a vital role in the structural study of Banach spaces and appeared for the first time in the book by Banach [4]. A systematic study of this concept was taken up by Grothendieck [26] in the year 1955 who considered the approximation property, bounded approximation property, and the basis property. At present, we have several variants of this property such as metric approximation property, compact approximation property, strong approximation property, M. Gupta (B) · D. Baweja Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India e-mail: [email protected] D. Baweja e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 M. Ruzhansky et al. (eds.), Advances in Real and Complex Analysis with Applications, Trends in Mathematics, DOI 10.1007/978-981-10-4337-6_2
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p-approximation property, and ideal approximation property, cf. [8, 20–22, 34–36, 45, 50], etc. As the identity operator on the space is approximated by linear operators having simpler representation in the study of the approximation property, there are three standard tools for studying approximation property for spaces of holomorphic mappings: -products, linearization, and S-absolute decompositions. The notion of -products for locally convex spaces X and Y written as X Y introduced by L. Schwartz is defined as the space Le (Yc ; X ) of all continuous linear operators from Yc to X , endowed with the topology of uniform convergence on equicontinuous subsets of Y , where Yc is the topological dual of Y equipped with the topology of uniform convergence on compact subsets of Y . Using the method of -products, the study of the approximation property for spaces of holomorphic mappings was initiated by Aron and Schottenloher in their pioneer work [2] and was further carried out in [14–19, 27, 29, 41, 49]. Through linearization results, one identifies a given class of holomorphic functions defined on an open subset U of a Banach space E with values in a Banach space F, with the space of continuous linear mappings from a certain Banach space G to F, i.e., a holomorphic
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