Abstract
Using the notion of S ξ -strictly singular operators introduced by Androulakis, Dodos, Sirotkin and Troitsky, we define an ordinal index on the subspace of strictly singular operators between two separable Banach spaces. In our main result, we provide a sufficient condition implying that this index is bounded by ω 1. In particular, we apply this result to study operators on totally incomparable spaces, hereditarily indecomposable spaces and spaces with few operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. E. Alspach and S. A. Argyros, Complexity of weakly null sequences, Dissertationes Mathematicae (Rozprawy Matematyczne) 321 (1992), 44.
G. Androulakis and K. Beanland, Descriptive set theoretic methods applies to strictly singular and strictly cosingular operators, Quaestiones Mathematicae 31 (2008), 151–161.
G. Androulakis, P. Dodos, G. Sirotkin and V. G. Troitsky, Classes of strictly singular operators and their products, Israel Journal of Mathematics 169 (2009), 221–250.
S. A. Argyros and I. Deliyanni, Examples of asymptotic l 1 Banach spaces, Transactions of the American Mathematical Society 349 (1997), 973–995.
S. A. Argyros and P. Dodos, Genericity and amalgamation of classes of Banach spaces, Advances in Mathematics 209 (2007), 666–748.
S. A. Argyros, G. Godefroy and H. P. Rosenthal, Descriptive set theory and Banach spaces, in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1007–1069.
S. A. Argyros and A. Manoussakis, A sequentially unconditional Banach space with few operators, Proceedings of the London Mathematical Society. Third Series 91 (2005), 789–818.
S. A. Argyros and S. Todorcevic, Ramsey Methods in Analysis, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2005.
S. A. Argyros and A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Mathematica 159 (2003), 1–32. Dedicated to Professor Aleksander Peńlczyński on the occasion of his 70th birthday.
S. A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Memoirs of the American Mathematical Society 170 (2004), vi+114.
S. Banach, Théorie des óperations linéaires, Chelsea Publishing Co., New York, 1955.
J. Bourgain, On separable Banach spaces, universal for all separable reflexive spaces, Proceedings of the American Mathematical Society 79 (1980), 241–246.
P. Dodos, On classes of banach spaces admitting ’small’ universal spaces, Transactions of the American Mathematical Society, Transactions of the American Mathematical Society 361 (2009), 6407–6428.
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, Journal of the American Mathematical Society 6 (1993), 851–874.
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer-Verlag, New York, 1995.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Springer-Verlag, Berlin, 1977; Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
E. Odell, Ordinal indices in Banach spaces, Extracta Mathematicae 19 (2004), 93–125.
A. Pełczyński, Universal bases, Studia Mathematica 32 (1969), 247–268.
W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Mathematica 30 (1968), 53–61.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beanland, K. An ordinal indexing on the space of strictly singular operators. Isr. J. Math. 182, 47–59 (2011). https://doi.org/10.1007/s11856-011-0023-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-011-0023-7