Pilar Rueda

Ideals of Homogeneous Polynomials

Given a surjective ideal of operators, we undertake a new general procedure to construct an ideal of polynomials. The relation with the ideal of polynomials obtained by the well-known method of composition is established. 2010 Mathematics Subject Classification: Primary 46G20; Secondary 46B20, 46G25.

Weakly uniformly continuous holomorphic functions and the approximation property

Abstract We study the approximation property for spaces of Frechet and Gâteaux holomorphic functions which are weakly uniformly continuous on bounded sets. We show when U is a balanced open subset of a Baire or barrelled metrizable locally convex space, E , that the space of holomorphic functions which are weakly uniformly continuous on U -bounded sets has the approximation property if and only if the strong dual of E , E ′ b , has the approximation property. We also characterise the approximation property for these spaces of vector-valued holomorphic functions in terms of the tensor product of the corresponding space of scalar-valued holomorphic functions and the range space.

p-Compact holomorphic mappings

Following Sinha and Karn [9], a relatively compact subset K of a Banach space E is said to be p-compact if for some sequence (xn) ∈ lp(E), K ⊂ {Σnaxxn | (an) ∈ Bℓ′p}. In [4], Delgado, Oja, Piñeiro, and Serrano investigated the p-approximation property, in which one only requires finite rank approximation of the identity on p-compact subsets. We investigate analogous concepts here for the case of holomorphic mappings between Banach spaces, introducing the space of p-compact holomorphic mappings (cf. [1]). A number of problems related to such holomorphic mappings are discussed.Resumen.Según Sinha y Karn [9], un subconjunto relativamente compacto K de un espacio de Banach E es p-compacto si para alguna sucesión (xn) ∈ lp(E), K ⊂ {Σnaxxn | (an) ∈ Bℓ′p}. En [4], Delgado, Oja, Piñeiro, y Serrano investigaron la propiedad de p-aproximación, en la que sólo se requiere aproximaciones en conjuntos p-compactos. En este trabajo investigamos conceptos análogos para aplicaciones holomorfas entre espacios de Banach, introduciendo el espacio de aplicaciones holomorfas p-compactas (cf. [1]). También presentamos una colección de problemas relacionados con dichas aplicaciones.

Dominated polynomials on infinite dimensional spaces

The aim of this paper is to prove a stronger version of a conjecture posed earlier on the existence of nondominated scalar-valued m-homogeneous polynomials, m > 3, on arbitrary infinite dimensional Banach spaces.

ON THE BANACH-DIEUDONNÉ THEOREM FOR SPACES OF HOLOMORPHIC FUNCTIONS

Abstract In this paper we show that a Banach-Dieudonne type theorem for the space of entire functions of bounded type on a Banach space only holds in the finite dimensional case. We also study if this result holds in the setting of Frechet spaces.

On Pietsch measures for summing operators and dominated polynomials

Let be the canonical map, where is a regular Borel probability measure on the closed unit ball of the dual of a Banach space endowed with the weak* topology. This paper has a twofold purpose: (i) to study when -summing linear operators/-dominated homogeneous polynomials on have a Pietsch measure for which the canonical map is injective; (ii) to show how these results can be used to correct the proofs of some results of the authors concerning Pietsch-type factorization of dominated polynomials.

Factorization of strongly (p, σ)-continuous multilinear operators

We introduce the new ideal of strongly-continuous linear operators in order to study the adjoints of the -absolutely continuous linear operators. Starting from this ideal we build a new multi-ideal by using the composition method. We prove the corresponding Pietsch domination theorem and we present a representation of this multi-ideal by a tensor norm. A factorization theorem characterizing the corresponding multi-ideal – which is also new for the linear case – is given. When applied to the case of the Cohen strongly -summing operators, this result gives also a new factorization theorem.

Dominated Bilinear Forms and 2-homogeneous Polynomials

The main goal of this note is to establish a connection between the cotype of the Banach space X and the parameters r for which every 2-homogeneous polynomial on X is r-dominated. Let cotX be the infimum of the cotypes assumed by X and (cotX)* be its conjugate. The main result of this note asserts that if cotX > 2, then for every 1<= r < (cotX)* there exists a non-r-dominated 2-homogeneous polynomial on X.

Summability and estimates for polynomials and multilinear mappings

Abstract In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on l p spaces in fact hold true for mappings on arbitrary Banach spaces.

Isometries of weighted spaces of holomorphic functions on unbounded domains

We study isometries between weighted spaces of holomorphic functions on unbounded domains in ℂ n . We show that weighted spaces of holomorphic functions on unbounded domains may exhibit behaviour different from that observed on bounded domains. We calculate the isometries for specific weights on the complex plane and the right half-plane.