Operators on \(\mathcal L_\infty\)-spaces
type: Item
instance of: scholarly article
title: Operators on \(\mathcal L_\infty\)-spaces
zbMATH Open document ID: 1333.46015
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: In [Acta Math. 145, 155--176 (1980; Zbl 0466.46024)], \textit{J. Bourgain} and \textit{F. Delbaen} presented a method of constructing \(\mathscr{L}_\infty\)-spaces not containing \(c_0\) which was proved to be extremely efficient. They built spaces of two types; the first class, \(\mathfrak{X}\), consists of spaces with the Schur property, the second, \(\mathfrak{Y}\), contains spaces which are reflexively saturated. \textit{S. A. Argyros} and \textit{R. G. Haydon} [Acta Math. 206, No. 1, 1--54 (2011; Zbl 1223.46007)] combined the Bourgain-Delbaen construction with techniques of mixed Tsirelson space theory, obtaining important examples of spaces. Modelling a BD-space on a classical mixed Tsirelson space, they obtained an unconditionally saturated \(\mathscr{L}_\infty\)-space not containing \(\ell_p\), \(1\leq p<\infty\), or \(c_0\). On the other hand, using the saturation methods in the Bourgain-Delbaen framework, Argyros and Haydon [loc.\,cit.]\ solved the famous scalar-plus-compact problem: They constructed a hereditarily indecomposable (HI) space which is a predual of \(\ell_1\) with the following property: Every bounded operator on the space is a compact perturbation of a scalar multiple of the identity. The compact-plus-scalar property is not determined by the HI structure of predual spaces to \(\ell_1\), as was shown by \textit{M. Tarbard} [J. Lond. Math. Soc., II. Ser. 85, No. 3, 737--764 (2012; Zbl 1257.46013)], who built an HI space, predual of \(\ell_1\), admitting a strictly singular non-compact operator. Regarding spaces with more regular structure, the Bourgain-Delbaen method combined with the Tsirelson-type construction was applied in [\textit{I. Gasparis} et al., Serdica Math. J. 36, No. 2, 149--170 (2010; Zbl 1224.05495)] to construct BD-spaces saturated with \(\ell_r\), \(1<r<\infty\). One should also recall here another application of the aforementioned methods, the striking result of \textit{S. A. Argyros} et al. [J. Funct. Anal. 262, No. 3, 825-…
reviewed by: Anna Pelczar-Barwacz
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47L05
zbMATH DE Number: 6420283
zbMATH Keywords: bounded linear operators
zbMATH Keywords: \(\mathscr{L}_\infty\)-space
zbMATH Keywords: Bourgain-Delbaen construction
zbMATH Keywords: mixed Tsirelson spaces
zbMATH Keywords: space of operators
author: Kevin Beanland
author: Lon H. Mitchell
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.12.003
OpenAlex ID: W83377008
cites work: EMBEDDING $\ell_{\infty}$ INTO THE SPACE OF BOUNDED OPERATORS ON CERTAIN BANACH SPACES
cites work: Embedding uniformly convex spaces into spaces with very few operators
cites work: A hereditarily indecomposable \(\mathcal L_{\infty}\)-space that solves the scalar-plus-compact problem
cites work: Operators on the∞-spaces of Bourgain and Delbaen
cites work: A class of special L//infinity spaces
cites work: Q4298827
cites work: On Banach spaces with the Gelfand-Phillips property. III
cites work: The universality of \(\ell _{1}\) as a dual space
cites work: More (\ell_r) saturated (\mathcal{L}_\infty) spaces
cites work: Subspaces of the Bourgain-Delbaen space
cites work: Spaces of compact operators
cites work: Hereditarily indecomposable, separable \(\mathcal L_{\infty }\) Banach spaces with \(\ell _{1}\) dual having few but not very few operators
cites work: Spaces of diagonal operators
DOI: 10.1016/J.JMAA.2013.12.003
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description: scientific article; zbMATH DE number 6420283