## 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.## References

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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