M. C. Calderón-Moreno

Dense Linear Manifolds of Monsters

In this paper the new concept of totally omnipresent operators is introduced. These operators act on the space of holomorphic functions of a domain in the complex plane. The concept is more restrictive than that of strongly omnipresent operators, also introduced by the authors in an earlier work, and both of them are related to the existence of functions whose images under such operators exhibit an extremely wild behaviour near the boundary. Sufficient conditions for an operator to be totally omnipresent as well as several outstanding examples are provided. After extending a statement of the first author about the existence of large linear manifolds of hypercyclic vectors for a sequence of suitable continuous linear mappings, it is shown that there is a dense linear manifold of holomorphic monsters in the sense of Luh, so completing earlier nice results due to Luh and Grosse-Erdmann.

Universal Transforms of the Geometric Series under Generalized Riesz Methods

In this paper generalized Riesz methods (R, p, M) of summability are considered. We prove that, to each open set O ⊂ ℂ with adequate topological properties and to each sequence {n} ⊂ ℂ tending to infinity, we can associate a corresponding P-regular (R, p, M)-method so that the geometric series and a certain trigonometric series become universal in the sense that its (R, p, M)-transforms approximate any member of certain spaces of holomorphic functions or measurable functions.

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.

UNIVERSALITY AND SUMMABILITY OF TRIGONOMETRIC POLYNOMIALS AND TRIGONOMETRIC SERIES

In 1945 Mensov proved the existence of a so-called universal trigonometric series ∞ ν=0 {aν cos νt+ bν sin νt}

Universal Taylor series with maximal cluster sets

We link the overconvergence properties of certain Taylor series in the unit disk to the maximality of their cluster sets, so connecting outer wild behavior to inner wild behavior. Specifically, it is proved the existence of a dense linear manifold of holomorphic functions in the disk that are, except for zero, universal Taylor series in the sense of Nestoridis and, simultaneously, have maximal cluster sets along many curves tending to the boundary. Moreover, it is constructed a dense linear manifold of universal Taylor series having, for each boundary point, limit zero along some path which is tangent to the corresponding radius. Finally, it is proved the existence of a closed infinite dimensional manifold of holomorphic functions enjoying the two-fold wild behavior specified at the beginning. 2000 Mathematics Subject Classification: Primary 30B30. Secondary 30D40, 30E10, 47B38.

Strongly omnipresent operators: general conditions and applications to composition operators

This paper studies the concept of strongly omnipresent operators that was recently introduced by the first two authors. An operator T on the space H(G) of holomorphic functions on a complex domain G is called strongly omnipresent whenever the set of T–monsters is residual in H(G), and a T–monster is a function f such that Tf exhibits an extremely ‘wild’ behaviour near the boundary. We obtain sufficient conditions under which an operator is strongly omnipresent, in particular, we show that every onto linear operator is strongly omnipresent. Using these criteria we completely characterize strongly omnipresent composition and multiplication operators. 2000 Mathematics subject classification: primary 30E10; secondary 30H05, 46E10, 47B38.

Simultaneously maximal radial cluster sets

In this paper, we show that for a wide class of operators T--including infinite order differential operators, and multiplication and composition operators--acting on the space H(D) of holomorphic functions in the unit disk D, we have most functions f ∈ H(D) which enjoy the property that T f has maximal radial cluster set at any boundary point.

Holomorphic Differential Operators and Plane Sets with Dense Images

We prove in this paper that, given a nonempty open set G in the complex plane, a subset A of G which is not relatively compact and a holomorphic infinite order differential or antidiffearential operator T, then there are holomorphic functions ƒ on G such that the image of A under T ƒ is dense in the complex plane. This extends a recent result about a property of boundary behaviour exhibited by the derivative operator.We prove in this paper that, given a nonempty open set G in the complex plane, a subset A of G which is not relatively compact and a holomorphic infinite order differential or antidiffeärential operator T, then there are holomorphic functions ƒ on G such that the image of A under T ƒ is dense in the complex plane. This extends a recent result about a property of boundary behaviour exhibited by the derivative operator.

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

Abstract In this paper, a study of topological and algebraic properties of two families of functions from the unit interval I into the plane R 2 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 → R 2 whose images have the positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures.

Compositional universality in the N-dimensional ball

It is proved in this note that a sequence of automorphisms on the N-dimensional unit ball acts properly discontinuously if and only if its corresponding sequence of composition operators is universal on the Hardy space of such ball, and if and only if there exists a dense linear manifold of universal functions. Our result completes earlier ones by several authors.