The Role of the Spectrum in the Cyclic Behavior of Composition Operators

A bounded operator $T$ acting on a Hilbert space $\\mathcal H$ is called cyclic if there is a vector $x$ such that the linear span of the orbit $\\{T^n x: n \\geq 0 \\}$ is dense in $\\mathcal H$. If the scalar multiples of the orbit are dense, then $T$ is called supercyclic. Finally, if the orbit itself is dense, then $T$ is called hyper cyclic. We completely characterize the cyclicity, the supercyclicity and the hypercyclicity of scalar multiples of composition operators, whose symbols are linear fractional maps, acting on weighted Dirichlet spaces. Particular instances of these spaces are the Bergman space, the Hardy space, and the Dirichlet space.Thus, we complete earlier work on cyclicity of linear fractional composition operators on these spaces. In this way, we find exactly the spaces in which these composition operators fail to be cyclic, supercyclic or hyper cyclic. Consequently, we answer some open questions posed by Zorboska. In almost all the cases, the cut-off of cyclicity, supercyclicity or hypercyclicity of scalar multiples is determined by the spectrum. We will find that the Dirichlet space plays a critical role in the cut-off.