Block diagonal polynomials

Dimant, Verónica; Gonzalo, Raquel

doi:10.1090/S0002-9947-00-02735-5

Block diagonal polynomials
HTML articles powered by AMS MathViewer

by Verónica Dimant and Raquel Gonzalo PDF

Trans. Amer. Math. Soc. 353 (2001), 733-747 Request permission

Abstract:

In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of $\ell _\infty$ in spaces of polynomials and to the stabilization of polynomial algebras are given.

References

Alvaro Arias and Jeff D. Farmer, On the structure of tensor products of $l_p$-spaces, Pacific J. Math. 175 (1996), no. 1, 13–37. MR 1419470, DOI 10.2140/pjm.1996.175.13
Richard M. Aron and Josip Globevnik, Analytic functions on $c_0$, Rev. Mat. Univ. Complut. Madrid 2 (1989), no. suppl., 27–33. Congress on Functional Analysis (Madrid, 1988). MR 1057205
R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. MR 552473, DOI 10.1515/crll.1980.313.195
Jacek Bochnak and Józef Siciak, Polynomials and multilinear mappings in topological vector spaces, Studia Math. 39 (1971), 59–76. MR 313810, DOI 10.4064/sm-39-1-59-76
F. Bombal and M. Fernández-Unzueta, Polynomial convergence of sequences in Banach spaces, Rev. Acad. Cienc. Exact. Fís. Natur. Madrid, to appear.
C. Boyd and R. A. Ryan, Bounded weak continuity of homogeneous polynomials at the origin, Arch. Math. (Basel) 71 (1998), no. 3, 211–218. MR 1637369, DOI 10.1007/s000130050254
J. Castillo, R. García and R. Gonzalo, Banach spaces in which all multilinear forms are weakly sequentially continuous, Studia Math. 136, 2 (1999), 121-145.
Verónica Dimant and Seán Dineen, Banach subspaces of spaces of holomorphic functions and related topics, Math. Scand. 83 (1998), no. 1, 142–160. MR 1662092, DOI 10.7146/math.scand.a-13846
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
Seán Dineen, A Dvoretsky theorem for polynomials, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2817–2821. MR 1264807, DOI 10.1090/S0002-9939-1995-1264807-1
S. Dineen, “Complex analysis on Infinite Dimensional Spaces”, Springer Monographs in Mathematics, 1999.
Seán Dineen and Mikael Lindström, Spaces of homogeneous polynomials containing $c_0$ or $l^\infty$, Functional analysis (Trier, 1994) de Gruyter, Berlin, 1996, pp. 119–127. MR 1420442
Jeff Farmer and William B. Johnson, Polynomial Schur and polynomial Dunford-Pettis properties, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 95–105. MR 1209450, DOI 10.1090/conm/144/1209450
Raquel Gonzalo, Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces, J. Math. Anal. Appl. 202 (1996), no. 2, 379–397. MR 1406236, DOI 10.1006/jmaa.1996.0322
Raquel Gonzalo, Upper and lower estimates in Banach sequence spaces, Comment. Math. Univ. Carolin. 36 (1995), no. 4, 641–653. MR 1378687
R. Gonzalo, Smoothness and polynomials on Banach spaces, Doctoral Thesis, Universidad Complutense de Madrid, 1994.
Petr Hájek, Polynomial algebras on classical Banach spaces, Israel J. Math. 106 (1998), 209–220. MR 1656881, DOI 10.1007/BF02773469
Francisco L. Hernández and Vicente Peirats, Weighted sequence subspaces of Orlicz function spaces isomorphic to $l^p$, Arch. Math. (Basel) 50 (1988), no. 3, 270–280. MR 933923, DOI 10.1007/BF01187745
J. R. Holub, Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 (1970), 563–579. MR 279564, DOI 10.1090/S0002-9947-1970-0279564-2
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
A. S. Nemirovskiĭ, The polynomial approximation of functions on Hilbert space, Funkcional. Anal. i Priložen. 7 (1973), no. 4, 88–89 (Russian). MR 0632032
K. Sundaresan, Geometry of spaces of homogeneous polynomials on Banach lattices, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 571–586. MR 1116377, DOI 10.1090/dimacs/004/43