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Let φ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that φ(0) = 1, φ(z) 6≡ 1. Let V be the operator of integration in Lp(0, 1), 1 ≤ p ≤ ∞. Then φ(V ) is power bounded if and only if φ′(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of φ(V ) and subsequent differences between the powers. It is shown that φ(V ) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if φ′(0) < 0, at least in the polynomial case.

The power boundedness and resolvent conditions for functions of the classical Volterra operator