Holomorphy Types Research Articles

Dynamics of non-convolution operators and holomorphy types

In this article we study the hypercyclic behavior of non-convolution operators defined on spaces of analytic functions of different holomorphy types over Banach spaces. The operators in the family we analyze are a composition of differentiation and composition operators, and are extensions of operators in H(C) studied by Aron and Markose in 2004. The dynamics of this class of operators, in the context of one and several complex variables, was further investigated by many authors. It turns out that the situation is somewhat different and that some purely infinite dimensional difficulties appear. For example, in contrast to the several complex variable case, it may happen that the symbol of the composition operator has no fixed points and still, the operator is not hypercyclic. We also prove a Runge type theorem for holomorphy types on Banach spaces.

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.

Hypercyclicity of convolution operators on spaces of entire functions

In this paper we use Nachbin's holomorphy types to generalize some recent results concerning hypercyclic convolution operators on Fr\'echet spaces of entire functions of bounded type of infinitely many complex variables.

$$(q;r)$$ -Dominated holomorphic mappings

In this paper we introduce and explore the notion of \((q;r)\)-dominated homogeneous polynomials. Among other results, we show that this concept lead to an ideal of polynomials which is a global holomorphy type and thus we introduce a natural version of \(( q;r)\)-dominated holomorphic mappings.

On almost summing polynomials and multilinear mappings

In this article we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach polynomial ideal which is a holomorphy type and also coherent and compatible with the notion of almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho.

Holomorphy types and the Fourier-Borel transform between spaces of entire functions of a given type and order defined on Banach spaces

Let $E$ be a Banach space and $\Theta$ be a $\pi_{1}$-holomorphy type. The main purpose of this paper is to show that the Fourier-Borel transform is an algebraic isomorphism between the dual of the space ${\operatorname{Exp}}_{\Theta,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $\Theta$-type strictly less than $A$ and the space ${\operatorname{Exp}}_{\Theta^{\prime},0,(\lambda (k) A)^{-1}}^{k^{\prime}}(E^{\prime})$ of entire functions on $E^{\prime}$ of order $k^{\prime}$ and $\Theta^{\prime}$-type less than or equal to $(\lambda(k)A)^{-1}$. The same is proved for the dual of the space ${\operatorname{Exp}}_{\Theta,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $\Theta$-type less than or equal to $A$ and the space ${\operatorname{Exp}}_{\Theta^{\prime}, (\lambda(k)A)^{-1}}^{k^{\prime}}( E^{\prime})$ of entire functions on $E^{\prime}$ of order $k^{\prime}$ and $\Theta^{\prime}$-type strictly less than $(\lambda(k)A)^{-1}$. Moreover, the Fourier-Borel transform is proved to be a topological isomorphism in certain cases.

On p-compact mappings and the p-approximation property

The notion of p-compact sets arises naturally from Grothendieckʼs characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm κp). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ϵ-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the κp-approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010).

Holomorphy types and spaces of entire functions of bounded type on banach spaces

In this paper spaces of entire functions of Θ-holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we “construct an algorithm” to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange: Existence et approximation des equations aux derivees partielles et des equations des convolutions. Annales de l’Institute Fourier (Grenoble) VI, 1955/56, 271–355; C. Gupta: Convolution Operators and Holomorphic Mappings on a Banach Space, Seminaire d’Analyse Moderne, 2, Universite de Sherbrooke, Sherbrooke, 1969; M. Matos: Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations, IMECC-UNICAMP, 2007; and X. Mujica: Aplicacoes τ (p; q)-somantes e σ(p)-nucleares, Thesis, Universidade Estadual de Campinas, 2006.

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

Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences

In this work we study mappings f from an open subset A of a Banach space E into another Banach space F such that, once a ∈ A is fixed, for mixed ( s ; q ) -summable sequences ( x j ) j = 1 ∞ of elements of a fixed neighborhood of 0 in E, the sequence ( f ( a + x j ) − f ( a ) ) j = 1 ∞ is absolutely p-summable in F. In this case we say that f is ( p ; m ( s ; q ) ) -summing at a. Since for s = q the mixed ( s ; q ) -summable sequences are the weakly absolutely q-summable sequences, the ( p ; m ( q ; q ) ) -summing mappings at a are absolutely ( p ; q ) -summing mappings at a. The nonlinear absolutely summing mappings were studied by Matos (see [Math. Nachr. 258 (2003) 71–89]) in a recent paper, where one can also find the historical background for the theory of these mappings. When s = + ∞ , the mixed ( ∞ , q ) -summable sequences are the absolutely q-summable sequences. Hence the ( p ; m ( ∞ ; q ) ) -summing mappings at a are the regularly ( p ; q ) -summing mappings at a. These mappings were also studied in [Math. Nachr. 258 (2003) 71–89] and they were important to give a nice characterization of the absolutely ( p ; q ) -summing mappings at a. We show that for q < s < + ∞ the space of the ( p ; m ( s ; q ) ) -summing mappings at a are different from the spaces of the absolutely ( p ; q ) -summing mappings at a and different from the spaces of regularly ( p ; q ) -summing mappings at a. We prove a version of the Dvoretzky–Rogers theorem for n-homogeneous polynomials that are ( p ; m ( s ; q ) ) -summing at each point of E. We also show that the sequence of the spaces of such n-homogeneous polynomials, n ∈ N , gives a holomorphy type in the sense of Nachbin. For linear mappings we prove a theorem that gives another characterization of ( s ; q ) -mixing operators in terms of quotients of certain operators ideals.

Hypercyclic convolution operators on entire functions of Hilbert-Schmidt holomorphy type

A theorem due to G. Godefroy and J. Shapiro states that every continuous convolution operator, that is not just multiplication by a scalar (non-trivial), is hypercyclic on the space of entire functions in n variables endowed with the compact-open topology. We study the space of entire functions of Hilbert-Schmidt type H H (E) on a Hilbert space E. We characterize its continuous convolution operators and prove the following: Every continuous non-trivial convolution operator is hypercyclic on H H (E).

Holomorphy types for open subsets of a Banach space

Holomorphy types for open subsets of a Banach space

Holomorphy types on a Banach spaces

Holomorphy types on a Banach spaces