No abstract

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 falsefalse∥false(T−λfalse)−1∥⩽Cfalse(|λ|−1false)−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.

Let A and B be positive operators on a Banach lattice such that the commutator C = AB − B A is also positive. We study the size of the spectrum of C.

We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan [1]).