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Algebra descent spectrum of operators

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Abstract

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ1, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.

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Authors and Affiliations

  1. Département de Mathématiques et Informatique, Faculté des Sciences, B.P. 20, El Jadida, Morocco

    A. Haïly

  2. Departamento de Algebra Y Analisis Matematico, Universidad de Almeria, 04120, Almería, Spain

    A. Kaidi

  3. Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias, 18071, Granada, Spain

    A. Rodríguez Palacios

Authors
  1. A. Haïly
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  2. A. Kaidi
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  3. A. Rodríguez Palacios
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Correspondence to A. Haïly.

Additional information

The authors are partially supported by the projects I+D MCYT MTM2004-03882, MTM-2006-15546-C02-02, MTM2007-65959, with FEDER founds, AECI PCI A/4044/05, A/5037/06, and the Junta de Andalucia grants FQM-194, FQM-199, FQM-1215.

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Haïly, A., Kaidi, A. & Rodríguez Palacios, A. Algebra descent spectrum of operators. Isr. J. Math. 177, 349–368 (2010). https://doi.org/10.1007/s11856-010-0050-9

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  • DOI: https://doi.org/10.1007/s11856-010-0050-9

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