Operadores de convolução hipercíclicos definidos entre espaços de Fréchet

Abstract

In this work, we study some results of the hypercyclicity theory. We show the proof of the first known examples of hypercyclic operators and we prove that every convolution operator defined between the spaces of all holomorphic functions defined on C, which is not a scalar multiple of the identity, is hypercyclic. We still study recent results of hypercyclicity for convolution operators defined between certain Fr´echet spaces of holomorphic functions defined on a complex Banach space. We show that the passage of the complex case to the case of an infinite dimensional complex Banach space is not trivial and this requires the use of several tools of holomorphy in infinite dimension.