The Individual Ergodic Theorem for Contractions with Fixed Points

1980 ◽  
Vol 23 (1) ◽  
pp. 115-116 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Individual Ergodic Theorem ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position

Let (X, I, μ) be a σ-finite measure space and let T take Lp to Lp, p fixed, 1<p<∞,‖t‖p≤1. We shall say that the individual ergodic theorem holds for T if for any uniform sequence K1, k2,… (for the definition, see [2]) and for any f∊LP(X), the limitexists and is finite almost everywhere.

Akcoglu's Ergodic Theorem for Uniform Sequences

1980 ◽  
Vol 32 (4) ◽  
pp. 880-884
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Contraction ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Image Position

Let (X, F,) be a sigma-finite measure space. In what follows we assume p fixed, 1 < p < ∞ . Let T be a contraction of Lp(X, F, μ) (‖T‖,p ≦ 1). If ƒ ≧ 0 implies Tƒ ≧ 0 we will say that T is positive. In this paper we prove that if is a uniform sequence (see Section 2 for definition) and T is a positive contraction of Lp, thenexists and is finite almost everywhere for every ƒ ∊ Lp(X, F, μ).

The Individual Weighted Ergodic Theorem for Bounded Besicovitch Sequences

1982 ◽  
Vol 25 (4) ◽  
pp. 468-471
Author(s):  
James H. Olsen
Keyword(s):  
Linear Operator ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Complex Numbers ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position ◽  
Weighted Ergodic Theorem

AbstractLet (X, , μ) be a σ-finite measure space, p fixed, 1 < p < ∞, T a linear operator of Lp(X,μ), {αi} a sequence of complex numbers. Ifexists and is finite a.e. we say the individual weighted ergodic theorem holds for T with the weights {αi}In this paper we show that if {αi} is a bounded Besicovitch sequence and T is a Dunford-Schwartz operator (i.e.: ||T||1≤1, ||T||∞≤1) then the individual weighted ergodic theorem holds for T with the weights {αi}.

An Extrapolation Theorem for Contractions with Fixed Points

1981 ◽  
Vol 24 (2) ◽  
pp. 199-203 ◽  
Author(s):  
Ryotaro Sato
Keyword(s):  
Linear Operator ◽  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Extrapolation Theorem ◽  
Image Position

In [9] de la Torre proved that if is a finite measure space and T is a linear operator on a real for some fixed p, 1 < p < ∞ , such that ||T||P ≤ 1 and simultaneously ||T||∞ ≤ l, and also such that there exists with Th = h and h≠0 a.e., then the dominated ergodic theorem holds for T, i.e. for every we havede la Torre proved his result, by showing that the operator S, defined by Sf = (sgn h) - T(f • sgn h) for is positive, and by applying Akcoglu's theorem [1] to S.

A Dominated Ergodic Theorem for Contractions with Fixed Points

1977 ◽  
Vol 20 (1) ◽  
pp. 89-91 ◽  
Author(s):  
A. De La Torre
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Finite Measure Space

Let be a finite measure space, and let T be a contraction in real Lp(X). (i.e. T is linear and ||T||≤1). It is said that the Dominated Ergodic Theorem holds for T, if there exists a constant cp such that, if M(T)f(x) = supn 1/n then ||M(T)f||p ≤ cp ||f||p for every f in Lp.

A Simple Proof of the Maximal Ergodic Theorem

1976 ◽  
Vol 28 (5) ◽  
pp. 1073-1075 ◽  
Author(s):  
Alberto De La Torre
Keyword(s):  
Ergodic Theorem ◽  
Simple Proof ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Function ◽  
Linear Transformations ◽  
Finite Measure Space ◽  
Image Position

Let X be a σ-finite measure space and let Tk, k any integer, be a group of positive linear transformations in Lp(X) such thatwith C independent of / and k. From now on / will be a positive function in Lp(X) and we will use the following notation:

A Multiple Sequence Ergodic Theorem

1983 ◽  
Vol 26 (4) ◽  
pp. 493-497 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Multiple Sequence ◽  
Finite Measure Space ◽  
Image Position

AbstractLet be a σ-finite measure space, {T1, …, Tk} a set of linear operators of , some p, 1≤p≤∞.Ifexists a.e. for all f ∊ Lp, we say that the multiple sequence ergodic theorem holds for {T1, …, Tk}. If f≥0 implies Tf≥0, we say that T is positive. If there exists an operator S such that |Tf(x)|≥S |f|(x) a.e., we say that T is dominated by S. In this paper we prove that if T1, …, Tk are dominated by positive contractions of , p fixed, 1<p<∞, then the multiple sequence ergodic theorem holds for {T1, …, Tk}.

A Local Ergodic Theorem on Lp

1974 ◽  
Vol 26 (5) ◽  
pp. 1206-1216 ◽  
Author(s):  
J. R. Baxter ◽  
R. V. Chacon
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Semi Group ◽  
Almost Everywhere ◽  
Local Case ◽  
Finite Measure Space ◽  
Continuity Assumption ◽  
Local Ergodic Theorem ◽  
Image Position

Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case.Let (X, , μ) be a complete, σ-finite measure space. Let {Tt} be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For Tt positive, it was shown independently in [2] and [5] that1.1almost everywhere on X, for any f ∊ L1. The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.

Ratio and Stochastic Ergodic Theorems for Superadditive Processes

1979 ◽  
Vol 31 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Humphrey Fong
Keyword(s):  
Linear Operator ◽  
Measure Space ◽  
Measure Zero ◽  
Positive Linear Operator ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Finite Measure Space ◽  
Image Position

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

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