Orbits of operators commuting with the Volterra operator

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 ⩽ ∞ , 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 ϕ ( V ) , where ϕ 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 − c V , where c > 0 . The results are applied to the study of cyclic properties of ϕ ( V ) , where ϕ is a holomorphic function at 0.