Dynamics of non-convolution operators and holomorphy types

Abstract

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, wasfurther 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.

Summing multilinear operators and sequence classes

“…Coherence of multi-ideals and coincidence results are classical topics in the theory of multilinear operators (see [34,35,36,52,53,64,63]). They make the interplay between multiideals/polynomials ideals and holomorphy types (see [21,30,34,36]), and applications to linear dynamics have been found (recent applications can be found in [32,33,49,50]). In this section we provide applications of our construction to these issues.…”

Operadores multilineares somantes por blocos e definidos por transformações de sequências vetoriais

“…São esses temas que relacionam multi-ideais com ideais de polinômios e com tipos de holomorfia (veja [21,36,40,42]). Além disso, ideais coerentes têm sido usados frequentemente em dinâmica linear (veja aplicações recentes em [38,39,59,60]). Nesta seção apresentaremos algumas aplicações da nossa construção para esses temas.…”