Illinois Journal of Mathematics

Operators on asymptotic $\ell_p$ spaces which are not compact perturbations of a multiple of the identity

Kevin Beanland

Full-text: Open access

Enhanced PDF (249 KB)

Abstract

We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space to ensure the space admits an operator, which is not a compact perturbation of a multiple of the identity. These conditions imply the existence of strictly singular noncompact operators on the HI spaces constructed by G. Androulakis and the author and by Deliyanni and Manoussakis. Additionally, we show that under these same conditions on the space $X$, $\ell_\infty$ embeds isomorphically into the space of bounded linear operators on $X$.

Article information

Source
Illinois J. Math., Volume 52, Number 2 (2008), 515-532.

Dates
First available in Project Euclid: 23 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1248355347

Digital Object Identifier
doi:10.1215/ijm/1248355347

Mathematical Reviews number (MathSciNet)
MR2524649

Zentralblatt MATH identifier
1185.46005

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces

Citation

Beanland, Kevin. Operators on asymptotic $\ell_p$ spaces which are not compact perturbations of a multiple of the identity. Illinois J. Math. 52 (2008), no. 2, 515--532. doi:10.1215/ijm/1248355347. https://projecteuclid.org/euclid.ijm/1248355347


Export citation

References

  • D. Alspach and S. A. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992), 44.
    Mathematical Reviews (MathSciNet): MR1191024
    Zentralblatt MATH: 0787.46009
  • G. Androulakis, K. Beanland, S. J. Dilworth and F. Sanacory, Embedding $\ell_{\infty}$ into the space of all operators on certain Banach spaces, Bull. London Math. Soc. 38 (2006), 979–990.
    Mathematical Reviews (MathSciNet): MR2285251
    Digital Object Identifier: doi:10.1112/S0024609306018868
  • G. Androulakis and K. Beanland, A hereditarily indecomposable asymptotic $\ell_2$ Banach space, Glasg. Math. J. 48 (2006), 503–532.
    Mathematical Reviews (MathSciNet): MR2271380
    Digital Object Identifier: doi:10.1017/S0017089506003193
  • G. Androulakis, E. Odell, Th. Schlumprecht and N. Tomczak-Jaegermann, On the structure of the spreading models of a Banach space, Canad. J. Math. 57 (2005), 673–707.
    Mathematical Reviews (MathSciNet): MR2152935
    Zentralblatt MATH: 1090.46004
    Digital Object Identifier: doi:10.4153/CJM-2005-027-9
  • G. Androulakis and F. Sanacory, An extension of Schreier unconditionality, Positivity 12 (2008), 313–340.
    Mathematical Reviews (MathSciNet): MR2399001
    Zentralblatt MATH: 1156.46017
    Digital Object Identifier: doi:10.1007/s11117-007-2126-2
  • 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.
    Mathematical Reviews (MathSciNet): MR1843416
    Zentralblatt MATH: 1015.46007
    Digital Object Identifier: doi:10.1112/S0024610701002769
  • S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), 243–294.
    Mathematical Reviews (MathSciNet): MR1750954
    Zentralblatt MATH: 0956.46014
    Digital Object Identifier: 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), 2019–2058 (electronic).
    Mathematical Reviews (MathSciNet): MR1813606
    Digital Object Identifier: 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), 157–193.
    Mathematical Reviews (MathSciNet): MR1658551
    Zentralblatt MATH: 0942.46007
    Digital Object Identifier: doi:10.1007/BF02764008
  • S. A. Argyros and A. Manoussakis, A sequentially unconditional Banach space with few operators, Proc. London Math. Soc. (3) 91 (2005), 789–818.
    Mathematical Reviews (MathSciNet): MR2180463
    Digital Object Identifier: doi:10.1112/S0024611505015388
  • S. A. Argyros and S. Todorcevic, Ramsey methods in analysis, Advanced Courses in Mathematics, CRM Barcelona, Birkhauser Verlag, Basel, 2005.
    Mathematical Reviews (MathSciNet): MR2145246
    Zentralblatt MATH: 1092.46002
  • N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. (2) 60 (1954), 345–350.
    Mathematical Reviews (MathSciNet): MR0065807
    Zentralblatt MATH: 0056.11302
    Digital Object Identifier: doi:10.2307/1969637
  • I. Deliyanni and A. Manoussakis, Asymptotic $\ell _p$ hereditarily indecomposable Banach spaces, Illinois J. Math. 51 (2007), 767–803.
    Mathematical Reviews (MathSciNet): MR2379722
    Digital Object Identifier: doi:10.1215/ijm/1258131102
    Project Euclid: euclid.ijm/1258131102
  • M. Feder, On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 34 (1980), 196–205.
    Mathematical Reviews (MathSciNet): MR0575060
    Zentralblatt MATH: 0411.46009
    Digital Object Identifier: doi:10.1215/ijm/1256047715
    Project Euclid: euclid.ijm/1256047715
  • V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc. 29 (1997), 338–344.
    Mathematical Reviews (MathSciNet): MR1435570
    Zentralblatt MATH: 0899.46010
    Digital Object Identifier: doi:10.1112/S0024609396002718
  • I. Gasparis, Strictly singular non-compact operators on hereditarily indecomposable Banach spaces, Proc. Amer. Math. Soc. 131 (2003), 1181–1189 (electronic).
    Mathematical Reviews (MathSciNet): MR1948110
    Zentralblatt MATH: 1019.46006
    Digital Object Identifier: doi:10.1090/S0002-9939-02-06657-1
  • I. Gasparis, A continuum of totally incomparable hereditarily indecomposable Banach spaces, Studia Math. 151 (2002), 277–298.
    Mathematical Reviews (MathSciNet): MR1917838
    Zentralblatt MATH: 1004.46005
    Digital Object Identifier: doi:10.4064/sm151-3-6
  • I. Gasparis and D. H. Lueng, On the complemented subspaces of the Schreier spaces, Studia Math. 131 (2000), 273–300.
    Mathematical Reviews (MathSciNet): MR1784674
    Digital Object Identifier: doi:10.4064/sm-141-3-273-300
  • W. T. 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. 111–115.
    Mathematical Reviews (MathSciNet): MR1665582
  • W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874.
    Mathematical Reviews (MathSciNet): MR1201238
    Zentralblatt MATH: 0827.46008
    Digital Object Identifier: doi:10.1090/S0894-0347-1993-1201238-0
  • N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267–278.
    Mathematical Reviews (MathSciNet): MR0341154
    Zentralblatt MATH: 0266.47038
    Digital Object Identifier: doi:10.1007/BF01432152
  • J. Lindenstrauss, Some open problems in Banach space theory, Seminaire Choquet. Initiation A l'analysie 15 (1975–1976), Expose 18, 9 pp.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin, 1977.
    Mathematical Reviews (MathSciNet): MR0500056
    Zentralblatt MATH: 0362.46013
  • Th. Schlumprecht, How many operators exist on a Banach space? Trends in Banach spaces and operator theory, Contemp. Math., vol. 321, Amer. Math. Soc., Providence, RI, 2003, pp. 295–333.
    Mathematical Reviews (MathSciNet): MR1978824
    Zentralblatt MATH: 1038.46007

Duke University Press

Illinois Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more