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ON STRICTLY SINGULAR OPERATORS BETWEEN SEPARABLE BANACH SPACES

  • Kevin Beanland (a1) and Pandelis Dodos (a2)

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(p,Yp) is a co-analytic non-Borel subset of ℒ(p,Yp) yet every strictly singular operator T:pYp satisfies ϱ(T)≤2. This answers a question of Argyros.

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[1]Alspach, D. and Argyros, S. A., Complexity of weakly null sequences. Dissertationes Math. 321 (1992), 144.
[2]Amemiya, I. and Ito, T., Weakly null sequences in James spaces on trees. Kodai J. Math. 88 (1968), 3546.
[3]Androulakis, G. and Beanland, K., Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators. Quaest. Math. 31 (2008), 151161.
[4]Androulakis, G., Dodos, P., Sirotkin, G. and Troitsky, V. G., Classes of strictly singular operators and their products. Israel J. Math. 169 (2009), 221250.
[5]Argyros, S. A. and Dodos, P., Genericity and amalgamation of classes of Banach spaces. Adv. Math. 209 (2007), 666748.
[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).
[7]Argyros, S. A. and Todorčević, S., Ramsey Methods in Analysis (Advanced Courses in Mathematics: CRM Barcelona), Birkhäuser (Basel, 2005).
[8]Beanland, K., An ordinal indexing of the space of strictly singular operators, Israel J. Math. (to appear).
[9]Bossard, B., A coding of separable Banach spaces. Analytic and co-analytic families of Banach spaces. Fund. Math. 172 (2002), 117152.
[10]Bourgain, J., On separable Banach spaces, universal for all separable reflexive spaces. Proc. Amer. Math. Soc. 79 (1980), 241246.
[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), 12581268.
[12]Dodos, P., On classes of Banach spaces admitting “small” universal spaces. Trans. Amer. Math. Soc. 361 (2009), 64076428.
[13]Dodos, P., Banach Spaces and Descriptive Set Theory: Selected Topics (Lecture Notes in Mathematics 1993), Springer (Berlin, 2010).
[14]Dodos, P. and Ferenczi, V., Some strongly bounded classes of Banach spaces. Fund. Math. 193 (2007), 171179.
[15]Dodos, P. and Lopez-Abad, J., On unconditionally saturated Banach spaces. Studia Math. 188 (2008), 175191.
[16]Ferenczi, V., On the number of pairwise permutatively inequivalent basic sequences in a Banach space. J. Funct. Anal. 238 (2006), 353373.
[17]Gasparis, I., A dichotomy theorem for subsets of the powerset of the natural numbers. Proc. Amer. Math. Soc. 129 (2001), 759764.
[18]Gowers, W. T. and Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), 851874.
[19]James, R. C., A separable somewhat reflexive Banach space with non-separable dual. Bull. Amer. Math. Soc. 80 (1974), 738743.
[20]Kechris, A. S., Classical Descriptive Set Theory (Graduate Texts in Mathematics 156), Springer (Berlin, 1995).
[21]Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces Vol. I: Sequence Spaces (Ergebnisse 92), Springer (Berlin, 1977).
[22]Odell, E., Ordinal indices in Banach spaces. Extracta Math. 19 (2004), 93125.
[23]Popov, A. I., Schreier singular operators. Houston J. Math. 35 (2009), 209222.
[24]Ramsey, F. P., On a problem of formal logic. Proc. Lond. Math. Soc. 30 (1930), 264286.
[25]Szlenk, W., The non existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces. Studia Math. 30 (1968), 5361.
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ON STRICTLY SINGULAR OPERATORS BETWEEN SEPARABLE BANACH SPACES

  • Kevin Beanland (a1) and Pandelis Dodos (a2)

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