Super-Ergodic operators

Yahdi, M.

doi:10.1090/S0002-9939-06-08255-4

Super-Ergodic operators
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Proc. Amer. Math. Soc. 134 (2006), 2613-2620 Request permission

Abstract:

The aim of this work is to study operators naturally connected to Ergodic operators in infinite-dimensional Banach spaces, such as Uniform-Ergodic, Cesaro-bounded and Power-bounded operators, as well as stable and superstable operators. In particular, super-Ergodic operators are introduced and shown to be strictly between Ergodic and Uniform-Ergodic operators, and that any power bounded operator is super-Ergodic in a superreflexive space. New relationships between these operators are shown, others are proven to be optimal or can be ameliorated according to structural properties of the Banach space, such as the superreflexivity or with unconditional basis.

References

I. Assani, Sur les opérateurs à puissances bornées et le théorème ergodique ponctuel dans $L^p[0,1]$, $1<p<+\infty$, Canad. J. Math. 38 (1986), no. 4, 937–946 (French). MR 854147, DOI 10.4153/CJM-1986-046-6
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
R. Émilion, Mean-bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), no. 1, 1–14. MR 779737, DOI 10.1016/0022-1236(85)90037-0
W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann. of Math. (2) 156 (2002), no. 3, 797–833. MR 1954235, DOI 10.2307/3597282
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
Edgar R. Lorch, Means of iterated transformations in reflexive vector spaces, Bull. Amer. Math. Soc. 45 (1939), 945–947. MR 1460, DOI 10.1090/S0002-9904-1939-07122-X
Rainer Nagel and Frank Räbiger, Superstable operators on Banach spaces, Israel J. Math. 81 (1993), no. 1-2, 213–226. MR 1231188, DOI 10.1007/BF02761307
Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735
Mohammed Yahdi, The topological complexity of sets of convex differentiable functions, Rev. Mat. Complut. 11 (1998), no. 1, 79–91. MR 1634617, DOI 10.5209/rev_{R}EMA.1998.v11.n1.17300
M. Yahdi, The spectrum of a superstable operator and coanalytic families of operators, Illinois J. Math. 45 (2001), no. 1, 91–111. MR 1849987, DOI 10.1215/ijm/1258138256