Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-11T13:11:51.015Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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