Linear Kierst-Szpilrajn theorems

We prove in this paper the following result which extends in a somewhat ‘linear’ sense a theorem by Kierst and Szpilrajn and which holds on many ‘natural’ spaces of holomorphic functions in the open unit disk D: There exist a dense linear manifold and a closed infinite-dimensional linear manifold of holomorphic functions in D whose domain of holomorphy is D except for the null function. The existence of a dense linear manifold of noncontinuable functions is also shown in any domain for its full space of holomorphic functions.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). Españ

Backward Φ-shifts and universality

In this paper we consider spaces of sequences which are valued in a topological space E and study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz’s theorem on hypercyclicity of scalar multiples of the classical backward shift is extended.Plan Andaluz de Investigación (Junta de Andalucía

Holomorphic functions having large images under the action of differential operators

We prove in this note that, given a simply connected domain G in the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functions f on G such that the image of any open subset under the action of those operators on f is arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened.Dirección General de Enseñanza Superior (DGES). Españ

On universal entire functions with zero-free derivatives

We prove in this note a generalization of a theorem due to G. Herzog on zero-free universal entire functions. Specifically, it is shown that, if a nonnegative integer q and a nonconstant entire function Φ of subexponential type are given, then there is a residual set in the class of entire functions with zero-free derivatives of orders q and q + 1, such that every member of that set is universal with respect to Φ(D), where D is the differentiation operator.Dirección General de Investigación Científica y Técnica (DGICYT). Españ

Hypercyclic sequences of differential and antidifferential operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hypercyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.Dirección General de Investigación Científica y Técnica (DGICYT). Españ

Small entire functions with extremely fast growth

We prove in this note that, given α ∈ (0, 1/2), there exists a linear manifold M of entire functions satisfying that M is dense in the space of all entire functions such that limz→∞ exp(|z|α)f(j)(z) = 0 on any plane strip for every f ∈ M and for every derivation index j. Moreover, the growth index of each nonnull function of M is infinite with respect to any prefixed sequence of nonconstant entire functions.Dirección General de Investigación Científica y Técnica (DGICYT). Españ

Compositionally universal entire functions on the plane and the punctured plane

Necessary/sufficient conditions for a sequence of automorphisms of the complex plane to generate a sequence of composition operators that is universal on the punctured plane are provided. As a consequence, it is derived that only for translations and rotation-dilations there can be entire functions whose orbits present universality. Boundedness of these functions on unbounded sets as well as frequent universality on the whole plane are also analyzed.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e InnovaciónMinisterio de Ciencia y Tecnología (MCYT). Españ

Hypercyclic subspaces in Fréchet spaces

In this note, we show that every infinite-dimensional separable Fr´echet space admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes earlier work of several authors.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). Españ

Linear structure of the weighted holomorphic non-extendibility

In this paper, it is proved that, for any domain G of the complex plane, there exist an infinite-dimensional closed linear submanifold M1 and a dense linear submanifold M2 with maximal algebraic dimension in the space H(G) of holomorphic functions on G such that G is the domain of holomorphy of every nonzero member of f of M1 or M2 and, in addition, the growth of f near each boundary point is as fast as prescribed.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). Españ