Alfonso Montes-Rodríguez

Semi-Fredholm Theory: Hypercyclic and Supercyclic Subspaces

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Universal functions for composition operators

Let H(Ω) be the space of analytic functions on a complex region Ω. In this paper, we characterize the sequences {ϕ n } n≥1of automorphisms of Ω with the following property:There exists f∊ H(Ω) such that the orbit {f˚ ϕ n } n≥1is dense in H(Ω). Some known results are derived and some new others are obtained.

The role of the spectrum in the cyclic behavior of composition operators

Introduction and preliminaries Linear fractional maps with an interior fixed point Non elliptic automorphisms The parabolic non automorphism Supercyclic linear fractional composition operators Endnotes Bibliography.

Spectral theory and hypercyclic subspaces

A vector x in a Hilbert space H is called hypercyclic for a bounded operator T : H → H if the orbit {Tnx : n ≥ 1} is dense in H. Our main result states that if T satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for T . The converse is true even if T is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator T to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator T : f(z)→ f(z+1) defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.

The role of the angle in supercyclic behavior

Abstract A bounded operator T acting on a Hilbert space H is said to be supercyclic if there is a vector f∈ H such that the projective orbit {λT n f : n⩾0 and λ∈ C } is dense in H . We use a new method based on a very simple geometric idea that allows us to decide whether an operator is supercyclic or not. The method is applied to obtain the following result: A composition operator acting on the Hardy space whose inducing symbol is a parabolic linear-fractional map of the disk onto a proper subdisk is not supercyclic. This result finishes the characterization of the supercyclic behavior of composition operators induced by linear fractional maps and, thus, completes previous work of Bourdon and Shapiro.

Perron-Frobenius operators and the Klein-Gordon equation

For a smooth curve Gamma and a set Lambda in the plane R-2, let AC(Gamma; Lambda) be the space of finite Borel measures in the plane supported on Gamma, absolutely continuous with respect to arc le ...

A note on birkhoff open sets

We characterize the open subsets G of the complex plane and the sequences of automorphisms of G for which there exists a universal holomorphic function f with the following property: the sequence is dense in the space of the holomorphic functions on G.