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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compact factorization of differentiable mappings
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by Raffaella Cilia, Joaquín M. Gutiérrez and Giuseppe Saluzzo PDF
Proc. Amer. Math. Soc. 137 (2009), 1743-1752 Request permission

Abstract:

Results on factorization (through linear operators) of polynomials and holomorphic mappings between Banach spaces have been obtained in recent years by several authors. In the present paper, we obtain a factorization result for differentiable mappings through compact operators. Namely, we prove that a mapping $f:X\to Y$ between real Banach spaces is differentiable and its derivative $f’$ is a compact mapping with values in the space ${\mathcal K}(X,Y)$ of compact operators from $X$ into $Y$ if and only if $f$ may be written in the form $f=g\circ S$, where the intermediate space is normed, $S$ is a precompact operator, and $g$ is a Gâteaux differentiable mapping with some additional properties. We also show that if $f’$ is uniformly continuous on bounded sets and takes values in ${\mathcal K}(X,Y)$, then $f’$ is compact if and only if $f$ is weakly uniformly continuous on bounded sets.
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Additional Information
  • Raffaella Cilia
  • Affiliation: Dipartimento di Matematica, Facoltà di Scienze, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy
  • MR Author ID: 326112
  • Email: cilia@dmi.unict.it
  • Joaquín M. Gutiérrez
  • Affiliation: Departamento de Matemática Aplicada, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain
  • MR Author ID: 311216
  • Email: jgutierrez@etsii.upm.es
  • Giuseppe Saluzzo
  • Affiliation: Dipartimento di Matematica, Facoltà di Scienze, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy
  • Email: saluzzo@dmi.unict.it
  • Received by editor(s): June 10, 2008
  • Published electronically: November 10, 2008
  • Additional Notes: The first and third authors were supported in part by G.N.A.M.P.A., Italy
    The first and second authors were supported in part by Dirección General de Investigación, MTM 2006–03531 (Spain)
  • Communicated by: Nigel J. Kalton
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1743-1752
  • MSC (2000): Primary 46G05; Secondary 47B10
  • DOI: https://doi.org/10.1090/S0002-9939-08-09716-5
  • MathSciNet review: 2470833