Operators with dense images everywhere

In this paper, the authors introduce the dense-image operators T as those with a wild behaviour near of the boundary of a domain G, via certain subsets. The relationship with other kinds of operators with wild behaviour is studied, proving that the new concept generalizes the earlier of omnipresent, but there is no good relationship with the strongly omnipresent operators. We obtain, among other results, that the following kinds of operators are dense-image: onto linear operators; operators with local dense range satisfying soft conditions; Volterra complex integral operators plus infinite order differential operators, multiplication operators. In addition, holomorphic selfmappings and entire functions generating dense-image right or left composition operators are completely characterized.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

Monsters in Hardy and Bergman spaces

A monster in the sense of Luh is a holomorphic function on a simply connected domain in the complex plane such that it and all its derivatives and antiderivatives exhibit an extremely wild behaviour near the boundary. In this paper the Hardy spaces Hp and the Bergman spaces Bp (1 ≤ p < ∞) on the unit disk are considered, and it is shown that there are no Luh-monsters in them. Nevertheless, it is proved that T-monsters (as introduced by the authors in an earlier work) can be found in each of these spaces for any finite order linear differential operator T.Plan Andaluz de Investigación (Junta de Andalucía

A Seidel-Walsh theorem with linear differential operators

Assume that {Sn}∞1 is a sequence of automorphisms of the open unit disk D and that {Tn}∞1 is a sequence of linear differential operators with constant coefficients, both of them satisfying suitable conditions. We prove that for certain spaces X of holomorphic functions in the open unit disk, the set of functions f ∈ X such that {(Tnf) ◦ Sn : n ∈ N} is dense in H(D) is residual in X. This extends the Seidel-Walsh theorem together with some subsequent results.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

Large algebras of singular functions vanishing on prescribed sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.Comment: 11 page

Hypercyclic algebras for D-multiples of convolution operators

It is shown in this short note the existence, for each nonzero member of the ideal of D-multiples of convolution operators acting on the space of entire functions, of a scalar multiple of it supporting a hypercyclic algebra.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Boundary-chaotic behaviour of continuous functions under the action of operators

In this paper we introduce two classes of operators on spaces of continuous functions with values in F-spaces under the action of which many functions behave chaotically near the boundary. Several examples, including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property, are given. This new theory extends one recently developed on spaces of holomorphic functions.Plan Andaluz de Investigación (Junta de Andalucía

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.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

Holomorphic T-monsters and strongly omnipresent operators

Assume that G is a nonempty open subset of the complex plane and that T is an operator on the linear space of holomorphic functions in G, endowed with the compact-open topology. In this paper we introduce the notions of strongly omnipresent operator and of T-monster, which are related to the wild behaviour of certain holomorphic functions near the boundary of G. T-monsters extend a concept introduced by W. Luh and K.-G. Grosse-Erdmann. After showing that T is strongly omnipresent if and only if the set of T-monsters is residual, it is proved in this paper that certain kinds of infinite order differential and antidifferential operators are strongly omnipresent, which improves some earlier nice results due to the mentioned authors.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

Two hyperbolic Schwarz lemmas

In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin, hence extending a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterized.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

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 each sequence {Pn} ⊂ 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.Plan Andaluz de Investigación (Junta de Andalucía