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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inclusion theorems for absolutely summing holomorphic mappings
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by Heinz Junek, Mário C. Matos and Daniel Pellegrino PDF
Proc. Amer. Math. Soc. 136 (2008), 3983-3991 Request permission

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 $n\geq 2,$ 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 $1\leq p\leq q\leq 2$ and $n\geq 2$, 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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3983-3991
  • MSC (2000): Primary 46B15; Secondary 46G25
  • DOI: https://doi.org/10.1090/S0002-9939-08-09394-5
  • MathSciNet review: 2425739