Factorization of injective ideals by interpolation

AbstractWe construct a factorization of certain multilinear mappings through linear operators belonging to closed, injective operator ideals using interpolation technique. An extension of the duality theorem for interpolation spaces is also obtained

Absolutely continuous multilinear operators

We introduce the new class of the (p;p1,...,pm; s)-absolutely continuous operators, that is defined using a summability property that provides the multilinear version of the (p, s)-absolutely continuous operators. We give an analogue of the Pietsch domination theorem and a multilinear version of the associated factorization theorem that holds for (p, s)-absolutely continuous operators, obtaining in this way a rich factorization theory. We present also a tensor norm which represents this multi-ideal by trace duality. As an application, we show that (p; p1, . . . , pm; s)-absolutely continuous multilinear operators are compact under some requirements. Applications to factorization of linear maps on Banach function spaces through interpolation spaces are also given.EL-Hadj Dahia acknowledges with thanks the support of the Ministere de l'Enseignament Superieur et de la Recherche Scientifique (Algeria) under grant 555/PGRS/C.U.K.M. (2011) for a short term stage. D. Achour acknowledges with thanks the support of the Ministere de l'Enseignament Superieur et de la Recherche Scientifique (Algeria) under project PNR 8-U28-6543. E.A. Sanchez acknowledges with thanks the support of the Ministerio de Economia y Connpetitividad (Spain) under project #MTM2009-14483-C02-02.Dahia, E.; Achour, D.; Sánchez Pérez, EA. (2013). Absolutely continuous multilinear operators. Journal of Mathematical Analysis and Applications. 397:205-224. https://doi.org/10.1016/j.jmaa.2012.07.034S20522439

On types of polynomials and holomorphic functions on Banach spaces

In 1966 L. Nachbin introduced the notion of a holomorphy type to consider certain types of polynomials (f.i. compact, nuclear, absolutely summing) in a uniform way [7,8].Holomorphy types with special properties were studied by S. Dineen in 1971 (cf. [4]).Using the well developped theory of linear operator ideals ([9]) various methods of the construction of holomorphy types were presented in [1]. These methods work also in the case of p-normed and quasinormed ideals. After introducing the basic notions the factorization method will be studied here in more details.The main result will be the Theorem 5.1.Of special interest are multilinear operators of type $\mathcal L (\mathcal Ip)$ where $\mathcal Ip$ denotes the usual Schatten class of linear operators in Hilbert spaces.These multilinear operators can be characterized by the summablity of their eigenvalues or some other sort of associated sequences of reals (Proposition 4.1).The results will be applied to multilinear operators defined by kernels or convolutions and to holomorphic functions of ideal type

Holomorphy types and ideals of multilinear mappings

We explore a condition under which the ideal of polynomials generated by an ideal of multilinear mappings between Banach spaces is a global holomorphy type. After some examples and applications, this condition is studied in its own right. A final section provides applications to the ideals formed by multilinear mappings and polynomials which are absolutely (p; q)-summing at every point