Operators on \(\mathcal L_\infty\)-spaces
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--849 (2012; Zbl 1241.46003)] stating that any separable uniformly convex space can be embedded in an \(\mathscr{L}_\infty\)-space with the scalar-plus-compact property. The authors of the present paper address the question of separability of the space of bounded operators on a space of Bourgain-Delbaen type. The space of operators on the Argyros-Haydon HI space, as well as on any other space with scalar-plus-compact property, is separable. The same holds true in the case of the Tarbard space. On the other hand, \textit{G. Emmanuele} [J. Math. Pures Appl., IX. Sér. 72, No. 3, 327--333 (1993; Zbl 0830.46011)] proved that the space of bounded operators on any space in the class \(\mathfrak{X}\), while non-separable, does not contain a copy of \(\ell_\infty\). In the present paper, a criterion for non-separability of the space of operators on a BD-space is given, satisfied in particular by the spaces in the class \(\mathfrak{Y}\), the unconditionally saturated BD-space of Argyros-Haydon and the Gasparis-Papadiamantis-Zisimopoulou space. The criterion is stated in terms of properties of an index set \(\Gamma\) defining the corresponding BD-space \(X(\Gamma)\) and yields that certain permutations of the basis of \(X(\Gamma)\) induce an uncountable family of pairwise separated bounded linear operators.
scientific article; zbMATH DE number 6420283
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