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 epsilon H such that the projective orbit {lambdaT(n)f: ngreater than or equal to0 and lambda epsilon 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.