Operators with an ergodic power

ArticleOriginal scientific text

Title

Authors ¹, ², ³

Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
Departamento de Matemáticas, Universidad de Cantabria, 39071 Santander, Spain
UFR de Mathématiques, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France

We prove that if some power of an operator is ergodic, then the operator itself is ergodic. The converse is not true.

T. Bermúdez, M. González and A. Martinón, On the poles of the local resolvent, Math. Nachr. 193 (1998), 19-26.
T. Bermúdez, M. González and M. Mbekhta, Local ergodic theorems, Extracta Math. 13 (1997), 243-248.
T. Bermúdez and A. Martinón, On Neumann operators, J. Math. Anal. Appl. 200 (1996), 698-707.
L. Burlando, A generalization of the uniform ergodic theorem to poles of arbitrary order, Studia Math. 122 (1997), 75-98.
I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, 1968.
D. Drissi, On a theorem of Gelfand and its local generalizations, Studia Math. 123 (1997), 185-194.
N. Dunford, Spectral theory I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217.
U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, Berlin, 1985.
K. B. Laursen and M. Mbekhta, Operators with finite chain length and the ergodic theorem, Proc. Amer. Math. Soc. 123 (1995), 3443-3448.
M. Mbekhta et J. Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158.
M. Radjabalipour, Decomposable operators, Bull. Iranian Math. Soc. 9 (1978), 1L-49L.
R. Sine, A note on the ergodic properties of homeomorphisms, Proc. Amer. Math. Soc. 57 (1976), 169-172.
A. C. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley, New York, 1980.
H.-D. Wacker, Über die Verallgemeinerung eines Ergodensatzes von Dunford, Arch. Math. (Basel) 44 (1985), 539-546.