Polynomials on Banach spaces with unconditional bases

Grecu, Bogdan; Ryan, Raymond

doi:10.1090/S0002-9939-04-07738-X

Polynomials on Banach spaces with unconditional bases
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by Bogdan C. Grecu and Raymond A. Ryan PDF

Proc. Amer. Math. Soc. 133 (2005), 1083-1091 Request permission

Abstract:

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

References

Andreas Defant, Juan Carlos Díaz, Domingo García, and Manuel Maestre, Unconditional basis and Gordon-Lewis constants for spaces of polynomials, J. Funct. Anal. 181 (2001), no. 1, 119–145. MR 1818112, DOI 10.1006/jfan.2000.3702
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281–1286. MR 147881, DOI 10.2140/pjm.1961.11.1281
S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68. MR 270118, DOI 10.4064/sm-34-1-43-67
Mário C. Matos, On holomorphy in Banach spaces and absolute convergence of Fourier series, Portugal. Math. 45 (1988), no. 4, 429–450. MR 982911
Mário C. Matos and Leopoldo Nachbin, Reinhardt domains of holomorphy in Banach spaces, Adv. Math. 92 (1992), no. 2, 266–278. MR 1155467, DOI 10.1016/0001-8708(92)90066-T
Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. MR 1888309, DOI 10.1007/978-1-4471-3903-4
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6