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Proceedings of the American Mathematical Society

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

 

 

Compact factorization of differentiable mappings


Authors: Raffaella Cilia, Joaquín M. Gutiérrez and Giuseppe Saluzzo
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
Published electronically: November 10, 2008
MathSciNet review: 2470833
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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
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
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

DOI: https://doi.org/10.1090/S0002-9939-08-09716-5
Keywords: Fr\'echet differentiable mapping, weakly continuous function, factorization, compact operator
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.