In this paper an improvement of the generalization---obtained in a previous article [Bu1] by the author---of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Indeed, two bounded linear operators T and A are constructed: 1/(n^2) T^n converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and 1/(n^3) \sum_{k=0}^{n-1} A^k converges uniformly, the sum of the range of 1-A and the kernel of (1-A)^2 being closed. Nevertheless, 1 is a pole of the resolvent of neither T nor A.
Generalizations of Cesàro means and poles of the resolvent
BURLANDO, LAURA
2004-01-01
Abstract
In this paper an improvement of the generalization---obtained in a previous article [Bu1] by the author---of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Indeed, two bounded linear operators T and A are constructed: 1/(n^2) T^n converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and 1/(n^3) \sum_{k=0}^{n-1} A^k converges uniformly, the sum of the range of 1-A and the kernel of (1-A)^2 being closed. Nevertheless, 1 is a pole of the resolvent of neither T nor A.
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