Orbits of operators commuting with the Volterra operator

摘要

Asymptotic estimates of the norms of orbits of certain operators that commute with the classical Volterra operator V acting on L-P[0,1], with 1 <= p <= infinity, are obtained. The results apply not only to the Riemann-Liouville operator V-r and to I + V-r with r > 0, but also to operators of the form phi (V), where phi is a holomorphic function at zero. The method to obtain the estimates is based on the fact that the Riemann-Liouville operator as well as the Volterra operator can be related to the Levin-Pfluger theory of holomorphic functions of completely regular growth. Different methods, such as the Denjoy-Carleman theorem, are needed to analyze the behavior of the orbits of I - cV, where c > 0. The results are applied to the study of cyclic properties of phi (V), where phi is a holomorphic function at 0. (c) 2007 Elsevier Masson SAS. All rights reserved.