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THE POSITION OF $\mathcal{K}(X,Y)$ IN $\mathcal{L}(X,Y)$

Published online by Cambridge University Press:  13 August 2013

DANIELE PUGLISI
DANIELE PUGLISI*
Affiliation:
Department of Mathematics and Computer Sciences, University of Catania, Catania 95125, Italy e-mail: dpuglisi@dmi.unict.it
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Abstract

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In this paper we investigate the nature of family of pairs of separable Banach spaces (X, Y) such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$. It is proved that the family of pairs (X,Y) of separable Banach spaces such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$ is not Borel, endowed with the Effros–Borel structure.

Keywords

Primary 46B20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 409 - 417
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Albiac, F. and Kalton, N. J., Topics in Banach space theory, Graduate Texts in Mathematics, vol 233 (Springer, New York, NY, 2006).Google Scholar
2.Argyros, S. A. and Haydon, R. G., A hereditarily indecomposable $\mathcal{L}_\infty$-space that solves the scalar-plus-compact problem, Acta Math. 206 (1) (2011), 1–54.Google Scholar
3.Arterburn, D. and Whitley, R. J., Projections in the space of bounded linear operators, Pacific J. Math. 15 (1965), 739–746.Google Scholar
4.Bossard, B., A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fund. Math. 172 (2) (2002), 117–152.Google Scholar
5.Bourgain, J., New classes of $\mathcal{L}_p$-spaces, Lecture Notes in Mathematics, vol. 889 (Springer-Verlag, Berlin, Germany, 1981).Google Scholar
6.Bourgain, J. and Delbaen, F., A class of special $\mathcal{L}_\infty$ spaces, Acta Math. 145 (3–4) (1980), 155–176.Google Scholar
7.Delpech, S., A short proof of Pitt's compactness theorem, Proc. Amer. Math. Soc. 137 (4) (2009), 1371–1372.CrossRefGoogle Scholar
8.Diestel, J., Geometry of Banach spaces: selected topics, Lecture Notes in Mathematics, vol. 485 (Springer-Verlag, Berlin, Germany, 1975).CrossRefGoogle Scholar
9.Dodos, P., Banach spaces and descriptive set theory: selected topics, Lecture Notes in Mathematics, vol. 1993, (Springer-Verlag, Berlin, Germany, 2010).Google Scholar
10.Emmanuele, G., A remark on the containment of c 0 in spaces of compact operators, Math. Proc. Camb. Philos. Soc. 111 (2) (1992), 331–335.Google Scholar
11.Emmanuele, G., Answer to a question by M. Feder about $\mathcal{K}(X,Y)$, Rev. Mat. Univ. Complut. Madrid 6 (2) (1993), 263–266.Google Scholar
12.Freeman, D., Odell, E. and Schlumprecht, Th., The universality of ℓ1 as a dual space, Math. Ann. 351 (1) (2011), 149–186.Google Scholar
13.Kalton, N. J., Spaces of compact operators, Math. Ann. 208 (1974), 267–278.Google Scholar
14.Kalton, N. J. and Werner, D., Property (M), M-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178.Google Scholar
15.Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156 (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
16.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, Lecture Notes in Mathematics, vol. 338 (Springer-Verlag, Berlin, Germany, 1973).Google Scholar
17.Pelczynski, A., Universal bases, Studia Math. 32 (1969), 247–268.Google Scholar
18.Thorp, E., Projections onto the subspace of compact operators, Pacific J. Math. 10 (1960), 693–696.Google Scholar
19.Tong, A. E. and Wilken, D. R., The uncomplemented subspace K(E,F), Studia Math. 37 (1971), 227–236.Google Scholar