Let V denote the classical Volterra operator. In this work, sharp estimates of the norm of (I − V)n acting on Lp [0, 1], for 1 ⩽ p ⩽ ∞, are obtained. As a consequence, I − V acting on Lp [0, 1], with 1 ⩽ p ⩽ ∞, is power bounded if and only if p = 2. Thus the Volterra operator characterizes when Lp [0, 1] is a Hilbert space. By means of sharp estimates of the L1-norm of the nth partial sums of the generating function of the Laguerre polynomials on the unit circle, it is also proved that I − V is uniformly Kreiss bounded on the spaces Lp [0,1], for 1 ⩽ p ⩽ ∞. A bounded linear operator T on a Banach space is said to be Kreiss bounded if there is a constant C > 0 such that ∥ ( T − λ ) − 1 ∥ ⩽ C ( | λ | − 1 ) − 1 for ∣ λ ∣ > 1. If the same upper estimate holds for each of the partial sums of the resolvent, then T is said to be uniformly Kreiss bounded. This is, for instance, true for power bounded operators. For finite-dimensional Banach spaces, Kreiss' Matrix Theorem asserts that Kreiss boundedness is equivalent to T being power bounded. Thus, in the infinite-dimensional setting, even a much stronger property than Kreiss boundedness still does not imply power boundedness. It is also shown that, for general operators, uniform Abel boundedness characterizes Cesàro boundedness and, as a consequence, uniform Kreiss boundedness is characterized in terms of a Cesàro type boundedness of order 1. 2000 Mathematics Subject Classification 47B38, 47G10.