Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지

Multiple Weakly Summing Multilinear Mappings and Polynomials

  • Kim, Sung Guen (Department of Mathematics, Kyungpook National University)
  • Received : 2007.04.20
  • Published : 2007.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce and study a new class containing absolutely summing multilinear mappings and polynomials, which we call multiple weakly summing multilinear mappings and polynomials. We investigate some interesting properties about multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials defined on Banach spaces: In particular, we prove a kind of Dvoretzky-Rogers' Theorem and an ideal property for multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials. We also prove that the Aron-Berner extensions of multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials are also multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing.

Keywords

References

  1. R. Aron and P. Berner, A Hahn-Banach extension theorem for analytic functions, Bull. Soc. Math. France, 106(1978), 3-24.
  2. F. Bombal, D. Perez-Garcia, and I. Villanueva, Multilinear extensions of Grothendieck's theorem, Quart. J. Math., 55(2004), 441-450. https://doi.org/10.1093/qmath/hah017
  3. Y.S. Choi, S.G. Kim, Y. Melendez, and A. Tonge, Estimates for absolutely summing norms of polynomials and multilinear maps, Quart. J. Math., 52 (2001), 1-12. https://doi.org/10.1093/qjmath/52.1.1
  4. A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc., 106 (1989), 351-356. https://doi.org/10.1090/S0002-9939-1989-0947313-8
  5. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland, Amsterdam, 1993.
  6. J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.
  7. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London, 1999.
  8. M. Matos, The Dvoretzky-Rogers theorem for polynomial and multilinear mappings, in "45o. Seminario Brasileiro de Analise, Florianopolis, 1997", pp. 849-856.