Some approximation properties and nuclear operators in spaces of analytical functions
type: Item
instance of: scholarly article
title: Some approximation properties and nuclear operators in spaces of analytical functions
zbMATH Open document ID: 1404.46017
author: Sten Kaijser
author: Oleg Reinov
published in: Advances in Operator Theory
publication date: 2018-10-02T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aot/1537408976
review text: Recall that a Banach space \(X\) has the approximation property (\textit{AP}) if the identity operator in \(X\) can be approximated by finite rank operators in the topology of uniform convergence on compact sets. Since every compact set is contained in the closed absolutely convex hull of a sequence which converges to zero, the \textit{AP} can be reformulated as follows: A Banach space \(X\) has the AP if, for every sequence \((x_n)_n\subset X\) with \(\|x_n\| \longrightarrow0\) as \(n \to \infty\) and \(\varepsilon >0\), there exists a finite rank operator \(R\) in \(X\) such that \(\sup_{n \in \mathbb N}\|x_n-Rx_n\|<\varepsilon\). The authors take this last form of the \textit{AP} and introduce a weak version of the \textit{AP}: Denote by \(\mathcal C\) the set of all real positive sequences tending to \(\infty\). Then, for a sequence \(c=(c_n)_n \in \mathcal C\), a Banach space \(X\) has the approximation property with respect to \(c\) (\(AP_{[c]}\)) if, for every \(\varepsilon >0\) and any sequence \((x_n)_n\) in \(X\) with \(\|x_n\|\leq c_n^{-1}\), there exists a finite rank operator \(R\) in \(X\) such that \(\|Rx_n-x_n\| \leq \varepsilon\) for every \(n\). Also, if \(\mathcal C_0\) is a subset of \(\mathcal C\), \(X\) has the approximation property respect to \(\mathcal C_0\) \((AP_{[\mathcal C_0]})\) if it has the \(AP_{[c]}\) for all \(c \in \mathcal C_0\). These type of approximation properties generalize some \textit{AP}'s considered in different works. For instance, in [\textit{J. Bourgain} and \textit{O. Reinov}, Math. Nachr. 122, 19--27 (1985; Zbl 0584.46039)], it was shown that \(H^{\infty}\) (the space of bounded analytic functions in the unit disk) has the \(AP_{[\log(n)]}\) and, if \(\ell_p^{-1}=\{(c_n)_n \in \mathcal C : (c_n^{-1})_n \in \ell_p\}\), then the \(AP_{\ell_p^{-1}}\) coincide with the \(AP_s\) considered, among others, in [\textit{O. I. Reinov}, J. Math. Anal. Appl. 415, No. 2, 816--824 (2014; Zbl 1323.47020)], where \(\frac 1s=2-\frac…
reviewed by: Pablo Turco
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 6946454
zbMATH Keywords: nuclear operator
zbMATH Keywords: tensor product
zbMATH Keywords: approximation property
zbMATH Keywords: space of bounded analytical functions
MaRDI profile type: Publication
OpenAlex ID: W2889711481
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DOI: 10.15352/AOT.1805-1360
Wikidata QID: Q129212666
description: scientific article; zbMATH DE number 6946454