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ON STRICTLY SINGULAR OPERATORS BETWEEN SEPARABLE BANACH SPACES

Part of: Set theory Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  22 June 2010

Kevin Beanland  and
Pandelis Dodos
Kevin Beanland
Affiliation:
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, U.S.A. (email: kbeanland@vcu.edu)
Pandelis Dodos
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece (email: pdodos@math.ntua.gr)
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Abstract

Let X and Y be separable Banach spaces and denote by 𝒮𝒮(X,Y ) the subset of ℒ(X,Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set 𝒮𝒮(X,Y ). Our main results are summarized as follows. Firstly, we define a new rank r𝒮 on 𝒮𝒮(X,Y ). We show that r𝒮 is a co-analytic rank and that it dominates the rank ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math.169 (2009), 221–250]. Secondly, for every 1≤p<+∞, we construct a Banach space Yp with an unconditional basis such that 𝒮𝒮(ℓp,Yp) is a co-analytic non-Borel subset of ℒ(ℓp,Yp) yet every strictly singular operator T:ℓp→Yp satisfies ϱ(T)≤2. This answers a question of Argyros.

MSC classification

Secondary: 46B28: Spaces of operators; tensor products; approximation properties 03E15: Descriptive set theory
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 285 - 304
Copyright
Copyright © University College London 2010

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References

[1]Alspach, D. and Argyros, S. A., Complexity of weakly null sequences. Dissertationes Math. 321 (1992), 1–44.Google Scholar
[2]Amemiya, I. and Ito, T., Weakly null sequences in James spaces on trees. Kodai J. Math. 88 (1968), 35–46.Google Scholar
[3]Androulakis, G. and Beanland, K., Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators. Quaest. Math. 31 (2008), 151–161.Google Scholar
[4]Androulakis, G., Dodos, P., Sirotkin, G. and Troitsky, V. G., Classes of strictly singular operators and their products. Israel J. Math. 169 (2009), 221–250.Google Scholar
[5]Argyros, S. A. and Dodos, P., Genericity and amalgamation of classes of Banach spaces. Adv. Math. 209 (2007), 666–748.Google Scholar
[6]Argyros, S. A., Godefroy, G. and Rosenthal, H. P., Descriptive Set Theory and Banach Spaces (Handbook of the Geometry of Banach Spaces 2) (eds Johnson, W. B. and Lindenstrauss, J.), Elsevier (Amsterdam, 2003).Google Scholar
[7]Argyros, S. A. and Todorčević, S., Ramsey Methods in Analysis (Advanced Courses in Mathematics: CRM Barcelona), Birkhäuser (Basel, 2005).Google Scholar
[8]Beanland, K., An ordinal indexing of the space of strictly singular operators, Israel J. Math. (to appear).Google Scholar
[9]Bossard, B., A coding of separable Banach spaces. Analytic and co-analytic families of Banach spaces. Fund. Math. 172 (2002), 117–152.Google Scholar
[10]Bourgain, J., On separable Banach spaces, universal for all separable reflexive spaces. Proc. Amer. Math. Soc. 79 (1980), 241–246.Google Scholar
[11]Chalendar, I., Fricain, E., Popov, A. I., Timotin, D. and Troitsky, V. G., Finitely strictly singular operators between James spaces. J. Funct. Anal. 256 (2009), 1258–1268.Google Scholar
[12]Dodos, P., On classes of Banach spaces admitting “small” universal spaces. Trans. Amer. Math. Soc. 361 (2009), 6407–6428.Google Scholar
[13]Dodos, P., Banach Spaces and Descriptive Set Theory: Selected Topics (Lecture Notes in Mathematics 1993), Springer (Berlin, 2010).Google Scholar
[14]Dodos, P. and Ferenczi, V., Some strongly bounded classes of Banach spaces. Fund. Math. 193 (2007), 171–179.Google Scholar
[15]Dodos, P. and Lopez-Abad, J., On unconditionally saturated Banach spaces. Studia Math. 188 (2008), 175–191.Google Scholar
[16]Ferenczi, V., On the number of pairwise permutatively inequivalent basic sequences in a Banach space. J. Funct. Anal. 238 (2006), 353–373.Google Scholar
[17]Gasparis, I., A dichotomy theorem for subsets of the powerset of the natural numbers. Proc. Amer. Math. Soc. 129 (2001), 759–764.CrossRefGoogle Scholar
[18]Gowers, W. T. and Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), 851–874.Google Scholar
[19]James, R. C., A separable somewhat reflexive Banach space with non-separable dual. Bull. Amer. Math. Soc. 80 (1974), 738–743.CrossRefGoogle Scholar
[20]Kechris, A. S., Classical Descriptive Set Theory (Graduate Texts in Mathematics 156), Springer (Berlin, 1995).Google Scholar
[21]Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces Vol. I: Sequence Spaces (Ergebnisse 92), Springer (Berlin, 1977).Google Scholar
[22]Odell, E., Ordinal indices in Banach spaces. Extracta Math. 19 (2004), 93–125.Google Scholar
[23]Popov, A. I., Schreier singular operators. Houston J. Math. 35 (2009), 209–222.Google Scholar
[24]Ramsey, F. P., On a problem of formal logic. Proc. Lond. Math. Soc. 30 (1930), 264–286.Google Scholar
[25]Szlenk, W., The non existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces. Studia Math. 30 (1968), 53–61.Google Scholar