Operator-Theoretical Treatment of Markoff's Process and Mean Ergodic Theorem

Yosidå, Kôsaku; Kakutani, Shizuo

doi:10.2307/1968993

Cited by 160 publications

(101 citation statements)

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“…Thus various authors-notably F. Riesz [28] and Yosida and Kakutani [33]-extended the mean ergodic theorem to an abstract theorem asserting the convergence to a fix point of the means Tnx = (n + l)~122oT'x, where T is a linear transformation of a Banach space E into itself. Alaoglu and Birkhoff [l] then replaced the iterates (T'~) by a semi-group G of linear transformations and showed that convergence of certain general means of transforms of an element x of E is equivalent to the existence and uniqueness of a fix point y in the closed convex hull of the orbit of x under G. The persistence of the customary countability and uniform boundedness restrictions on G in their work, however, severely limits the generality.…”

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confidence: 99%

“…This remark covers Hubert space and, more generally, the spaces 2> and /* (p> 1). In the spaces L and / Riesz [28], Birkhoff [l], Kakutani [33] and others observed that a lattice-theoretically bounded set of functions-that is, [x|a(0 Sx(t) =b(t) identically, a, b(E.L or /]-is conditionally weakly compact. In a Lebesgue space L(S) a general criterion for weak compactness (Riesz [28], Dunford and Pettis [12]) yields the conditional weak compactness of Tnx if T is of the classical form Tx(t) = x(<pt), where 0 is a one-to-one measure-preserving transformation of 5 into itself (4).…”

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confidence: 99%

“…Conditions on the operators F of G sufficient to insure the ergodicity of every x in E are of interest. Yosida and Kakutani [33] found the special (bounded) case of Example 1 in which F is quasicompact fundamental in their abstract treatment of Markoff processes. [September Similarly, we obtain Tx = TxTa = TxU.…”

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Abstract ergodic theorems and weak almost periodic functions

Eberlein¹

1949

Trans. Amer. Math. Soc.

298

234

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Although the individual ergodic theorem of G. D. Birkhoff is sharper than the mean ergodic theorem of J. von Neumann, it was soon evident that the measure-theoretic formulation obscured the greater generality of the latter result. Thus various authors-notably F. Riesz [28] and Yosida and Kakutani[33]-extended the mean ergodic theorem to an abstract theorem asserting the convergence to a fix point of the means Tnx = (n + l)~122oT'x, where T is a linear transformation of a Banach space E into itself. Alaoglu and Birkhoff [l] then replaced the iterates (T'~) by a semi-group G of linear transformations and showed that convergence of certain general means of transforms of an element x of E is equivalent to the existence and uniqueness of a fix point y in the closed convex hull of the orbit of x under G. The persistence of the customary countability and uniform boundedness restrictions on G in their work, however, severely limits the generality.A fresh abstraction is thus required, not only to subsume present results in a sharper and more transparent form, but to extend the domain of the ergodicity phenomena.In Part I we study a semi-group G of linear transformations operating on a space £, G ordinarily being restricted by an "ergodicity" condition of the weakest type. We derive criteria for the validity of a mean ergodic theorem in an arbitrary locally-convex linear topological space £. Specializing £ to a Banach space E, we obtain not only standard theorems as obvious corollaries of the general theory, but significantly new results. For example, we obtain a mean ergodic theorem for an arbitrary bounded Abelian semi-group G on the one hand, and Fejér's theorem as an ergodic theorem for an unbounded semigroup on the other. The role played by weak compactness in E and (weak) quasi-compactness of the operators F of G is clarified, as well as the relation of ergodic theory to the mean value problem for generalized almost periodic functions, the relation being particularly simple in case the underlying group is Abelian.In Part II we consider a locally-compact Abelian group G acting as translations on the Banach algebra C(G) of bounded continuous complex-valued

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Abstract ergodic theorems and weak almost periodic functions

Eberlein¹

1949

Trans. Amer. Math. Soc.

298

234

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“…What is nowadays called the uniform ergodic theorem goes back to 1941 when Yosida and Kakutani published the study [52]. In this study they obtained conditions 1 on a bounded linear operator T in a Banach space B which are sufficient to provide that the averages n 01 P n01 j=0 T j converge in the uniform operator topology to an operator P in B :…”

Section: Introductionmentioning

confidence: 99%

“…They wrote (see [52] In the rest of the Yosida-Kakutani study the result has been well applied to some problems in the theory of Markov processes. A number of studies have followed.…”

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confidence: 99%

The Uniform Ergodic Theorem for Dynamical Systems

Peškir¹,

Weber²

1996

Convergence in Ergodic Theory and Probability

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Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann's mean and Birkhoff's pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves as a particular case Blum-DeHardt's theorem which offers the best known sufficient condition for the uniform law of large numbers in the independent case. Secondly it is shown that eventual tightness is necessary and sufficient. In this way a link with a weak convergence is obtained. Finally it is shown that the existence of some particular totally bounded pseudo-metrics is necessary and sufficient. The conditions derived are of Lipschitz type, while the method of proof relies upon a result of independent interest, called the uniform ergodic lemma. This result considerably extends Hopf-Yosida-Kakutani's maximal ergodic lemma to a form more suitable for examinations of the uniform convergence under consideration. From this lemma an inequality is also derived which extends the classical (weak) maximal ergodic inequality to the uniform case. In addition, a uniform approximation by means of a dense family of maps satisfying the uniform ergodic theorem in a trivial way is investigated, and a particular result of this type is established. This approach is in the spirit of the classical Hilbert space method for the mean ergodic theorem of von Neumann, and therefore from the ergodic theory point of view it could be seen as the natural one. After this, a simple characterization is obtained for the uniform convergence of moving averages. Finally, a counter-example is constructed for a symmetrization inequality in the stationary ergodic case. This inequality is known to be of vital importance to support the Vapnik-Chervonenkis random entropy approach in the independent case. Further developments in this direction are indicated.

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