×

zbMATH — the first resource for mathematics

Two new properties of ideals of polynomials and applications. (English) Zbl 1089.46027
An ideal of polynomials is a natural generalization of the notion of an operator ideal. The authors consider two properties of such ideals of polynomials.
Let \(\mathcal Q\) be an ideal of polynomials. 1. \({\mathcal Q}\) is closed under differentiation (\(\mathcal Q\) is cud) if \(\widehat{d} P(a)\in {\mathcal Q} (E;F)\) for every \(a\in E\) and every \(n\)-homogeneous continuous polynomial \(P\in {\mathcal Q }(^n E;F)\), 2. \(\mathcal Q\) is closed under scalar multiplication (\(\mathcal Q\) is csm) if \(\varphi P\in {\mathcal Q} (^{n+1} E;F)\) for every \(\varphi \in E'\) and \(P\in {\mathcal Q}(^n E;F)\) for all \(n\in \mathbb N\) and Banach spaces \(E\) and \(F\).
It is shown that the ideals \({\mathcal P}_{L_{(\mathcal I )}}\) and \({\mathcal P}_{[\mathcal I ]}\) share these properties. Here, \(\mathcal I\) is an operator ideal and \({\mathcal P}_{L_{(\mathcal I )}}, {\mathcal P}_{[\mathcal I ]}\) go back to ideals of multilinear mappings defined in [A. Pietsch, in: Operator algebras, ideals, and their applications in theoretical physics, Proc. int. Conf., Leipzig 1983, Teubner-Texte Math. 67, 185–199 (1984; Zbl 0562.47037)]. In this way, the authors give, among other results, a lot of new polynomial characterisations of \({\mathcal L}_{\infty}\)-spaces and spaces whose duals are isomorphic to \(\ell_1 (\Gamma )\), extending classical results.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aron, R.M.; Hervés, C.; Valdivia, M., Weakly continuous mappings on Banach spaces, J. funct. anal., 52, 189-204, (1983) · Zbl 0517.46019
[2] Botelho, G., Ideals of polynomials generated by weakly compact operators, Note mat., 24, (2005), in press
[3] Braunss, H., Ideale multilinearer abbildungen and räume holomorfer funktionen, ()
[4] Braunss, H.; Junek, H., Factorization of injective ideals by interpolation, J. math. anal. appl., 297, 740-750, (2004) · Zbl 1053.47059
[5] Cilia, R.; D’Anna, M.; Gutiérrez, J., Polynomial characterization of L∞-spaces, J. math. anal. appl., 275, 900-912, (2002) · Zbl 1031.46051
[6] Cilia, R.; D’Anna, M.; Gutiérrez, J., Polynomials on Banach spaces whose duals are isomorphic to ℓ_1(γ), Bull. austral. math. soc., 70, 117-124, (2004) · Zbl 1077.46042
[7] Diestel, J.; Uhl, J.J., ()
[8] Dineen, S., Complex analysis on infinite dimensional spaces, (1999), Springer-Verlag Providence, RI · Zbl 1034.46504
[9] González, M.; Gutiérrez, J.; González, M.; Gutiérrez, J., Injective factorization of holomorphic mappings, (), 1255-1256, See also erratum in
[10] Grothendieck, A., Résumé de la théorie métrique des produits tensoriels topologiques, Bol. soc. mat. São paulo, 8, 1-79, (1956) · Zbl 0074.32303
[11] Lewis, D.R.; Stegall, C., Banach spaces whose duals are isomorphic to ℓ_1(γ), J. funct. anal., 12, 177-187, (1973) · Zbl 0252.46021
[12] Lindenstrauss, J.; Pe&lstrokczyński, A., Absolutely summing operators in L_p spaces and their applications, Studia math., 29, 275-326, (1968)
[13] Matos, M.C., Absolutely summing holomorphic mappings, An. acad. brasil. ciênc., 68, 1-13, (1996) · Zbl 0854.46042
[14] Mujica, J., Complex analysis in Banach spaces, ()
[15] Pietsch, A., Operator ideals, (1980), North-Holland Amsterdam · Zbl 0399.47039
[16] Pietsch, A., Ideals of multilinear functionals (designs of a theory), (), 185-199
[17] Stegall, C.P., Banach spaces whose duals contain \scl_1(γ) with applications to the study of dual L1(μ) spaces, Trans. amer. math. soc., 176, 463-477, (1973) · Zbl 0259.46016
[18] Stegall, C.P.; Retherford, J.R., Fully nuclear and completely nuclear operators with applications to \scl_1 and L∞-spaces, Trans. amer. math. soc., 163, 457-492, (1972) · Zbl 0225.47012
[19] Wojtaszczyk, P, Banach spaces for analysts, () · Zbl 0858.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.