Strictly singular non-compact operators on hereditarily indecomposable Banach spaces
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- by I. Gasparis
- Proc. Amer. Math. Soc. 131 (2003), 1181-1189
- DOI: https://doi.org/10.1090/S0002-9939-02-06657-1
- Published electronically: July 26, 2002
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Abstract:
An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic $\ell _1$ Banach space. The construction of this operator relies on the existence of transfinite $c_0$-spreading models in the dual of the space.References
- Dale E. Alspach and Spiros Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44. MR 1191024
- G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125โ149. MR 1679593, DOI 10.1007/BF02775031
- 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), 1โ20.
- S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no.ย 3, 973โ995. MR 1390965, DOI 10.1090/S0002-9947-97-01774-1
- S. A. Argyros, I. Deliyanni, D. N. Kutzarova, and A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), no.ย 1, 43โ109. MR 1654174, DOI 10.1006/jfan.1998.3310
- S.A. Argyros, I. Deliyanni and A. Manoussakis, Distortion and spreading models in modified mixed Tsirelson spaces, preprint.
- S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), no.ย 2, 243โ294. MR 1750954, DOI 10.1090/S0894-0347-00-00325-8
- S. A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, Trans. Amer. Math. Soc. 353 (2001), no.ย 5, 2019โ2058. MR 1813606, DOI 10.1090/S0002-9947-01-02711-8
- S. A. Argyros, S. Mercourakis, and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157โ193. MR 1658551, DOI 10.1007/BF02764008
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611โ633. MR 16, DOI 10.2307/1968946
- V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc. 29 (1997), no.ย 3, 338โ344. MR 1435570, DOI 10.1112/S0024609396002718
- T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no $l_{p}$, Compositio Math. 29 (1974), 179โ190. MR 355537
- I. Gasparis and D. H. Leung, On the complemented subspaces of the Schreier spaces, Studia Math. 141 (2000), no.ย 3, 273โ300. MR 1784674, DOI 10.4064/sm-141-3-273-300
- I. Gasparis, A continuum of totally incomparable hereditarily indecomposable Banach spaces, submitted.
- W. Timothy Gowers, A remark about the scalar-plus-compact problem, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp.ย 111โ115. MR 1665582
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no.ย 4, 851โ874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
- Denka Kutzarova and Pei-Kee Lin, Remarks about Schlumprecht space, Proc. Amer. Math. Soc. 128 (2000), no.ย 7, 2059โ2068. MR 1654081, DOI 10.1090/S0002-9939-99-05248-X
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- V.I. Lomonosov, Invariant subspaces for operators commuting with compact operators, Functional Anal. Appl. 7 (1973), 213โ214.
- B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann, Asymptotic infinite-dimensional theory of Banach spaces, Geometric aspects of functional analysis (Israel, 1992โ1994) Oper. Theory Adv. Appl., vol. 77, Birkhรคuser, Basel, 1995, pp.ย 149โ175. MR 1353458
- E. Odell, On subspaces, Asymptotic Structure, and Distortion of Banach Spaces; Connections with Logic, Analysis and Logic, (C. Finet and C. Michaux, eds.) (2000), 301โ376 (to appear).
- Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no.ย 1-2, 81โ95. MR 1177333, DOI 10.1007/BF02782845
- J. Schreier, Ein Gegenbeispiel zur theorie der schwachen konvergenz, Studia Math. 2 (1930), 58โ62.
- B. S. Tsirelson, Not every Banach space contains $\ell _p$ or $c_0$, Funct. Anal. Appl. 8 (1974), p. 138โ141.
Bibliographic 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
- Received by editor(s): July 2, 2001
- Received by editor(s) in revised form: November 14, 2001
- Published electronically: July 26, 2002
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1181-1189
- MSC (2000): Primary 46B03; Secondary 06A07, 03E02
- DOI: https://doi.org/10.1090/S0002-9939-02-06657-1
- MathSciNet review: 1948110