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Operators with an integral reprsentation


Authors: Raffaella Cilia and Joaquín M. Gutiérrez
Journal: Proc. Amer. Math. Soc. 144 (2016), 5275-5290
MSC (2010): Primary 47B10; Secondary 47L20, 46B28, 46B03
DOI: https://doi.org/10.1090/proc/13249
Published electronically: July 28, 2016
MathSciNet review: 3556271
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Abstract: We introduce a fairly large class of bounded linear operators between Banach spaces which admit an integral representation. It turns out that an operator belongs to this class if and only if it factors through a $ C(K)$ space. As an application, we characterize Banach spaces containing no copy of $ c_0$, Banach spaces containing no complemented copy of $ \ell _1$, Grothendieck spaces, and $ \mathscr L_{\infty }$-spaces. We also study $ C(K)$-factorization and extension properties of absolutely continuous operators, giving a partial answer to a question raised in 1985 by H. Jarchow and U. Matter.


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Additional Information

Raffaella Cilia
Affiliation: Dipartimento di Matematica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy
Email: cilia@dmi.unict.it

Joaquín M. Gutiérrez
Affiliation: Departamento de Matemáticas del Área Industrial, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain
Email: jgutierrez@etsii.upm.es

DOI: https://doi.org/10.1090/proc/13249
Keywords: Integral representation, $\infty$-integral operators, extendible operators, factorization through $C(K)$ spaces, absolutely continuous operators
Received by editor(s): August 31, 2015
Received by editor(s) in revised form: February 19, 2016
Published electronically: July 28, 2016
Additional Notes: Both authors were supported in part by Dirección General de Investigación, MTM2015-65825-P Spain
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society