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

 

Inclusion theorems for absolutely summing holomorphic mappings


Authors: Heinz Junek, Mário C. Matos and Daniel Pellegrino
Journal: Proc. Amer. Math. Soc. 136 (2008), 3983-3991
MSC (2000): Primary 46B15; Secondary 46G25
Published electronically: June 11, 2008
MathSciNet review: 2425739
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Abstract: For linear operators, if $ 1\leq p\leq q<\infty,$ then every absolutely $ p$-summing operator is also absolutely $ q$-summing. On the other hand, it is well known that for $ n\geq2,$ there are no general ``inclusion theorems'' for absolutely summing $ n$-linear mappings or $ n$-homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely $ p$-summing and absolutely $ q$-summing linear operators coincide, and prove that for $ 1\leq p\leq q\leq2$ and $ n\geq2$, we have inclusion theorems for absolutely summing $ n$-linear mappings/$ n$-homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as $ p$ increases, the classes of absolutely $ p$-summing mappings becomes smaller.


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Additional Information

Heinz Junek
Affiliation: Institute of Mathematics, University of Potsdam, 14469, Potsdam, Germany
Email: junek@rz.uni-potsdam.de

Mário C. Matos
Affiliation: IMECC-UNICAMP, Caixa Postal 6065, Campinas, SP, Brazil
Email: matos@ime.unicamp.br

Daniel Pellegrino
Affiliation: Departamento de Matemática, UFPB, J. Pessoa, 58051-900, PB, Brazil
Email: dmpellegrino@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09394-5
PII: S 0002-9939(08)09394-5
Received by editor(s): December 29, 2006
Received by editor(s) in revised form: October 18, 2007
Published electronically: June 11, 2008
Additional Notes: The third author was supported by CNPq Grants 471054/2006-2 (Edital Universal) and 308084/2006-3
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.