Lineability and modes of convergence

In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in L1-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.Junta de AndalucíaMinisterio de Ciencia, Innovación y Universidades (MICINN). Españ

A dynamical characterization of sub-Arakelian subsets

We are going to characterize those sets which can be covered by an Arakelian set in terms of dynamical properties of entire functions via similarities. Moreover, if we consider the set of universal entire functions via similarities that are bounded on such a sub-Arakelian set, then it is shown that its algebraic size is as large as possible. As a consequence, we prove that the set of universal entire functions bounded on every (straight) line is algebraically large.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e Innovació

Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Algebraic structure of continuous, unbounded and integrable functions

In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in [0, +∞) and of the family of sequences of these functions converging to zero uniformly on compacta and in L1-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero.Junta de AndalucíaMinisterio de Economía y Competitivida

Maximal cluster sets along arbitrary curves

The existence of a dense linear manifold of holomorphic functions on a Jordan domain having except for zero maximal cluster set along any curve tending to the boundary with nontotal oscillation value set is shown.Plan Andaluz de Investigación (Junta de Andalucía

The set of space-filling curves: topological and algebraic structure

In this paper, a study of topological and algebraic properties of two families of functions from the unit interval I into the plane R2 is performed. The first family is the collection of all Peano curves, that is, of those continuous mappings onto the unit square. The second one is the bigger set of all space-filling curves, i.e. of those continuous functions I → R2 whose images have positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitivida

Large subspaces of compositionally universal functions with maximal cluster sets

Let (φn) be a sequence of holomorphic self-maps of a Jordan domain G in the complex plane. Under appropriate conditions on (φn), we construct an H(G)-dense linear manifold –as well as a closed infinite-dimensional linear manifold– all of whose non-zero functions have H(G)-dense orbits under the action of the sequence of composition operators associated to (φn). Simultaneously, these functions also present maximal cluster sets along each member of a large class of curves in G tending to the boundary.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e InnovaciónMinisterio de Ciencia y Tecnologí

Holomorphic operators generating dense images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that T f has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others.Key words and phrases: dense images, infinite dimensional closed linear spaces, non-relatively compact subsets, residual sets, differential operators, composition operators, multiplication operators.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superio

Cyclicity of coefficient multipliers: Linear structure

In this paper we characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic.Ministerio de Ciencia y TecnologíaPlan Andaluz de Investigación (Junta de Andalucía

Simultaneously maximal radial cluster sets

In this paper, we show that for a wide class of operators T –including infi- nite order differential operators, and multiplication and composition operators– acting on the space H(D) of holomorphic functions in the unit disk D, we have that most functions f ∈ H(D) enjoy the property that T f has maximal radial cluster set at any boundary point.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superio