Intertwining relations, commutativity and orbits

Abstract

The density of orbits and commutativity up to a factor of bounded linear operators have become of great interest for Operator Theorists during the last decades. This interest comes from the relationship that exists between the study of orbits and spectral properties of linear operators on Banach spaces and the Invariant Subspace Problem. As a consequence, the research on the Theory of Hyperclicity has increased considerably. In this manuscript, we characterize the hypercyclicity of the Cesàro means of higher-order on Banach spaces. We prove some sufficient conditions on the extended-spectrum of a bounded linear operator that guarantee its non convex-cyclicity. The notion of convex-cyclicity, was introduced by H. Rezaei in 2013 (see \cite{reza}). It is a sufficient condition for cyclicity and a necessary condition for hypercyclicity. We characterize the hypercyclicity of operators commuting up to a factor with the differentiation operator in the space of entire functions equipped with the topology of uniform convergence for compact sets. Our results are an extension of some of the most classical results related to the differentiation operator, that is, the ones of G. Godefroy and J. H. Shapiro \cite{Godefroy1991}, and R. Aron and D. Markose \cite{AronMarkose2004}. Next, we consider some particular operators, such as composition operators in weighted Hardy spaces. These operators have been studied intensely by several mathematicians in the Hardy space, see the recent of \cite{leon4}. Although we know fewer things about these operators in weighted Hardy spaces, we calculated the extended-spectrum of composition operators that are induced by a bilinear transformation that fixes an interior point of the unit disk and an exterior one of its closure. Namely, we treat the elliptic, loxodromic cases and a hyperbolic subcase. Finally, we continue to the study of the more general unbounded operators. After the paper of von Neumann \cite{von-Neumman-historic-Fuglede}, commutativity and intertwining relations of unbounded operators have been developed by many mathematicians. Among these mathematicians, we state Bent Fuglede whose Theorem was an improvement of the Spectral Theorem for Normal Operators. We show a new version of the Fuglede Theorem for unbounded normal operators.