On the uniform ergodic theorem in Banach spaces that do not contain duals

• Autorzy: Vladimir Fonf , Michael Lin , Alexander Rubinov
• Języki publikacji: EN

Abstrakty

EN

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X = {z ∈ X: sup_{n} ∥∑_{k=0}^{n} T^{k}z∥ < ∞}$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains on the (positive) integers are obtained.

Kategorie tematyczne

• 60J35: Transition functions, generators and resolvents
• 47A35: Ergodic theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 121
• Numer: 1
• Strony: 67-85

Daty

wydano

1996

otrzymano

1996-02-26

poprawiono

1996-04-26

Twórcy

autor

Vladimir Fonf

• Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

autor

Michael Lin

• Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

autor

Alexander Rubinov

• Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Bibliografia

• [BP] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
• [BoR] J. Bourgain and H. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), 149-188.
• [BrSu_1] A. Brunel et L. Sucheston, Sur quelques conditions équivalentes à la super-reflexivité dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. A 275 (1972), 993-994.
• [BrSu_2] A. Brunel et L. Sucheston, On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299.
• [BuW] P. Butzer and U. Westphal, Ein Operatorenkalkül für das approximationstheoretische Verhalten des Ergodensatzes im Mittel, in: Linear Operators and Approximation, P. Butzer, J.-P. Kahane and B. Sz.-Nagy (eds.), Birkhäuser, Basel, 1972, 102-114.
• [Da] W. J. Davis, Separable Banach spaces with only trivial isometries, Rev. Roumaine Math. Pures Appl. 16 (1971), 1051-1054.
• [D] M. Day, Normed Linear Spaces, 3rd ed., Springer, Berlin, 1973.
• [DuSc] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.
• [F_1] V. Fonf, One property of Lindenstrauss-Phelps spaces, Funct. Anal. Appl. 13 (1979), 66-67.
• [F_2] V. Fonf, Injections of Banach spaces with closed image of the unit ball, ibid. 19 (1985), 75-77.
• [F_3] V. Fonf, Dual subspaces and injections of Banach spaces, Ukrainian Math. J. 39 (1987), 285-289.
• [F_4] V. Fonf, On semi-embeddings and $G_δ$ embeddings of Banach spaces, Mat. Zametki 39 (1986), 550-561 (in Russian); English transl. in Math. Notes 39 (1986).
• [GHe] W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36, Providence, 1955.
• [Ho] S. Horowitz, Transition probabilities and contractions of $L_∞$, Z. Wahrsch. Verw. Gebiete 24 (1972), 263-274.
• [KoL] I. Kornfeld and M. Lin, Coboundaries of irreducible Markov operators on C(K), Israel J. Math., to appear.
• [K] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, Berlin, 1985.
• [KL] U. Krengel and M. Lin, On the range of the generator of a Markovian semi-group, Math. Z. 185 (1984), 553-565.
• [L_1] M. Lin, On quasi-compact Markov operators, Ann. of Probab. 2 (1974), 464-475.
• [L_2] M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337-340.
• [L_3] M. Lin, Quasi-compactness and uniform ergodicity of positive operators, Israel J. Math. 29 (1978), 309-311.
• [LS] M. Lin and R. Sine, Ergodic theory and the functional equation (I-T)x=y, J. Operator Theory 10 (1983), 153-166.
• [Loe] M. Loève, Probability Theory, 3rd ed., van Nostrand-Reinhold, New York, 1963.
• [Lo] H. Lotz, Uniform convergence of operators on $L^∞$ and similar spaces, Math. Z. 190 (1985), 207-220.
• [LPP] H. Lotz, N. Peck, and H. Porta, Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc. 22 (1979), 233-240.
• [M] A. Milyutin, Isomorphy of the spaces of continuous functions on metric compacta of cardinality $c_1$, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 2 (1966), 150-156 (in Russian).
• [P] A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Rozprawy Mat. 58 (1968).
• [R] A. Rubinov, Sublinear operators and their applications, Uspekhi Mat. Nauk 32 (4) (1977), 115-175 (in Russian); English transl.: Russian Math. Surveys 32 (4) (1977), 113-174.
• [Si] I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981.
• [Su] L. Sucheston, Problems, in: Probability in Banach Spaces, Lecture Notes in Math. 526, Springer, 1976, 285-290.
• [YH] K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46-66.
• [YKa] K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. (2) 42 (1941), 188-228.
• [W] R. Wittmann, Schwach irreduzible Markoff-Operatoren, Monatsh. Math. 105 (1988), 319-334.
• [Z] R. Zaharopol, Mean ergodicity of power-bounded operators in countably order complete Banach lattices, Math. Z. 192 (1986), 81-88.