On orbits of functions of the Volterra operator (Q371870)

The author studies the classical Volterra operator on the spaces \(L^p [0,1]\). Sharp estimates are obtained for the asymptotic behaviour of the iterates of \(I-V\) on the range of fractional (positive) powers of \(V\). This gives new proofs (for \(p=2\)) and interesting extensions of some recent results. The author's approach is based on \textit{D. Sarason}'s function model for the Volterra operator [J. Math. Anal. Appl. 12, 244--246 (1965; Zbl 0138.38801)]. A large class of the Ritt operators with the single spectrum \(\{1\}\) is provided. Fejér's asymptotic formula on the Laguerre polynomials is also involved.