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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On spaces of compact operators on $ C(K, X)$ spaces


Author: Elói Medina Galego
Journal: Proc. Amer. Math. Soc. 139 (2011), 1383-1386
MSC (2010): Primary 46B03; Secondary 46B25
Published electronically: August 23, 2010
MathSciNet review: 2748430
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Abstract: This paper concerns the spaces of compact operators $ {\mathcal K}(E, F)$, where $ E$ and $ F$ are Banach spaces $ C([1, \xi], X)$ of all continuous $ X$-valued functions defined on the interval of ordinals $ [1, \xi]$ and equipped with the supremun norm. We provide sufficient conditions on $ X$, $ Y$, $ \alpha$, $ \beta$, $ \xi$ and $ \eta$, with $ \omega \leq \alpha \leq \beta< \omega_{1}$ for the following equivalence:

(a)
$ {\mathcal K}(C([1, \xi], X), C([1, \alpha], Y))$ is isomorphic to $ {\mathcal K} (C([1, \eta], X), C([1, \beta], Y))$,

(b)
$ \beta< \alpha^{\omega}$.
In this way, we unify and extend results due to Bessaga and Pełczyński (1960) and C. Samuel (2009). Our result covers the case of the classical spaces $ X=l_{p}$ and $ Y=l_{q}$, with $ 1<p, q< \infty$.


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Additional Information

Elói Medina Galego
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
Email: eloi@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10544-0
PII: S 0002-9939(2010)10544-0
Keywords: Isomorphic classifications of spaces of compact operators
Received by editor(s): November 8, 2009
Received by editor(s) in revised form: April 9, 2010, and April 16, 2010
Published electronically: August 23, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.