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Strictly singular operators on $ L_p$ spaces and interpolation

Author(s): Francisco L. Hernández; Evgeny M. Semenov; Pedro Tradacete
Journal: Proc. Amer. Math. Soc. 138 (2010), 675-686.
MSC (2000): Primary 47B38; Secondary 47B07, 46B70
Posted: October 13, 2009
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Abstract: We study the class $ V_p$ of strictly singular non-compact operators on $ L_p$ spaces. This allows us to obtain interpolation results for strictly singular operators on $ L_p$ spaces. Given $ 1\leq p<q\leq\infty$, it is shown that if an operator $ T$ bounded on $ L_p$ and $ L_q$ is strictly singular on $ L_r$ for some $ p\leq r\leq q$, then it is compact on $ L_s$ for every $ p<s<q$.

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Additional Information:

Francisco L. Hernández
Affiliation: Departmento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain
Email: pacoh@mat.ucm.es

Evgeny M. Semenov
Affiliation: Department of Mathematics, Voronezh State University, Voronezh 394006, Russia
Email: semenov@func.vsu.ru

Pedro Tradacete
Affiliation: Departmento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain
Email: tradacete@mat.ucm.es

DOI: 10.1090/S0002-9939-09-10089-8
PII: S 0002-9939(09)10089-8
Keywords: Strictly singular operator, $L_p$ space, interpolation
Received by editor(s): February 18, 2009,
Received by editor(s) in revised form: June 18, 2009
Posted: October 13, 2009
Additional Notes: The first and third authors were partially supported by grants MICINN MTM2008-02652 and Santander/Complutense PR34/07-15837. The second author was partly supported by the Russian Fund. of Basic Research grants 08-01-00226-a and a Universidad Complutense grant. The third author was partially supported by grant MEC AP-2004-4841.
Communicated by: Nigel J. Kalton
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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