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Strictly singular non-compact operators on hereditarily indecomposable Banach spaces
Author(s):
I.
Gasparis
Journal:
Proc. Amer. Math. Soc.
131
(2003),
1181-1189.
MSC (2000):
Primary 46B03;
Secondary 06A07, 03E02
Posted:
July 26, 2002
MathSciNet review:
1948110
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Abstract:
An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic  Banach space. The construction of this operator relies on the existence of transfinite  -spreading models in the dual of the space.
References:
- 1.
- D. Alspach and S.A. Argyros, Complexity of weakly null sequences, Dissertationes Mathematicae 321 (1992), 1-44. MR 93j:46014
- 2.
- G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125-149. MR 2000f:46012
- 3.
- G. Androulakis and Th. Schlumprecht, Strictly singular, non-compact operators exist on the space of Gowers and Maurey, J. London Math. Soc. (2) 64 (2001), 1-20.
- 4.
- S.A. Argyros and I. Deliyanni, Examples of asymptotic
 Banach spaces, Trans. Amer. Math. Soc. 349 (1997), 973-995. MR 97f:46021 - 5.
- S.A. Argyros, I. Deliyanni, D.N. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), 43-109. MR 2000j:46031
- 6.
- S.A. Argyros, I. Deliyanni and A. Manoussakis, Distortion and spreading models in modified mixed Tsirelson spaces, preprint.
- 7.
- S.A. Argyros and V. Felouzis, Interpolating Hereditarily Indecomposable Banach Spaces, J. Amer. Math. Soc. 13 (2000), 243-294. MR 2002b:46021
- 8.
- S.A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, Trans. Amer. Math. Soc. 353 (2001), no. 5, 2019-2058. MR 2002b:46012
- 9.
- S.A. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157-193. MR 99m:46021
- 10.
- N. Aroszajn and K.T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. (2) 60 (1954), 345-350. MR 16:488b
- 11.
- V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc. 29 (1997), no.3, 338-344. MR 98b:47028
- 12.
- T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no
 , Compositio Math. 29 (1974), 179-190. MR 50:8011 - 13.
- I. Gasparis and D.H. Leung, On the complemented subspaces of the Schreier spaces, Studia Math. 141 (2000), no. 3, 273-300. MR 2001j:46009
- 14.
- I. Gasparis, A continuum of totally incomparable hereditarily indecomposable Banach spaces, submitted.
- 15.
- W.T. Gowers, A remark about the scalar-plus-compact problem, Convex geometric analysis (Berkeley, CA, 1996), 111-115. MR 99m:46015
- 16.
- W.T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. MR 94k:46021
- 17.
- D.N. Kutzarova and P.K. Lin, Remarks about Schlumprecht space, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2059-2068. MR 2000m:46031
- 18.
- J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Springer-Verlag, New York (1977). MR 58:17766
- 19.
- V.I. Lomonosov, Invariant subspaces for operators commuting with compact operators, Functional Anal. Appl. 7 (1973), 213-214.
- 20.
- B. Maurey, V.D. Milman and N. Tomczak-Jaegermann, Asymptotic infinite-dimensional theory of Banach spaces, Oper. Theory: Adv. Appl. 77 (1994), 149-175. MR 97g:46015
- 21.
- E. Odell, On subspaces, Asymptotic Structure, and Distortion of Banach Spaces; Connections with Logic, Analysis and Logic, (C. Finet and C. Michaux, eds.) (2000), 301-376 (to appear).
- 22.
- Th. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81-95. MR 93h:46023
- 23.
- J. Schreier, Ein Gegenbeispiel zur theorie der schwachen konvergenz, Studia Math. 2 (1930), 58-62.
- 24.
- B. S. Tsirelson, Not every Banach space contains
 or  , Funct. Anal. Appl. 8 (1974), p. 138-141.
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Additional Information:
I.
Gasparis
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Address at time of publication:
Department of Mathematics, University of Crete, Knossou Avenue, P.O. Box 2208, Herakleion, Crete 71409, Greece
Email:
ioagaspa@math.okstate.edu, ioagaspa@math.uch.gr
DOI:
10.1090/S0002-9939-02-06657-1
PII:
S 0002-9939(02)06657-1
Keywords:
Hereditarily Indecomposable space,
Tsirelson's space,
Schreier sets.
Received by editor(s):
July 2, 2001
Received by editor(s) in revised form:
November 14, 2001
Posted:
July 26, 2002
Communicated by:
N. Tomczak-Jaegermann
Copyright of article:
Copyright
2002,
American Mathematical Society
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