Czasopismo
Tytuł artykułu
Warianty tytułu
Języki publikacji
Abstrakty
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
67-85
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-02-26
poprawiono
1996-04-26
Twórcy
autor
- Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel, fonf@math.bgu.ac.il
autor
- Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel, lin@math.bgu.ac.il
autor
- Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel, rubinov@math.bgu.ac.il
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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv121i1p67bwm