Multiparameter admissible superadditive processes

Doğan Çömez

Displaying similar documents to “Multiparameter admissible superadditive processes”

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

Studia Mathematica

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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$ , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$ . For p = 1 we find that the...

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, Wojciech Kosek (2003)

Colloquium Mathematicae

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $| | T ⁿ | | \sim n^{1 / 4}$ ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty . I n t h e c l a s s o f p o s i t i v e o p e r a t o r s t h e s e e x a m p l e s a r e t h e b e s t p o s s i b l e i n t h e s e n s e t h a t f o r e v e r y s u c h o p e r a t o r T t h e r e e x i s t s a γ ₀ > 0 s u c h t h a t$ lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 $.$ A class of numerical sequences αₙ, intimately...

Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

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Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ (U)}_{n = 1}^{\infty}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ (U) = 1 / n \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) U^{k}$ . We show that this sequence ${ₙ (U)}_{n = 1}^{\infty}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral...

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

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We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of $ℤ ₊^{d}$ actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

Spectral theory and operator ergodic theory on super-reflexive Banach spaces

Earl Berkson (2010)

Studia Mathematica

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On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $s u p_{n \in ℕ, z \in} | | \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) k^{- 1} z^{k} U^{k} | | < \infty$ . (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with...

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

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Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$ . A loop $h : S^{1} \to G$ is called strictly ergodic if for some irrational number the associated skew product map $T : S^{1} \times Y \to S^{1} \times Y$ defined by $T (t, y) = (t + α; h (t) y)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...

Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek (2011)

Colloquium Mathematicae

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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty$ . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $l i m s u p | | T ⁿ | | / n^{1 - γ ₀} = 0 .$ In this note we construct an example of a nonpositive L¹ operator with the...

Vector-valued ergodic theorems for multiparameter additive processes

Ryotaro Sato (1999)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=( $u_{1}$ , ... , $u_{d})$ , $u_{i}$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_{1}$ ((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_{1}$ ((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_{1}$ ((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems...

Vector-valued ergodic theorems for multiparameter Additive processes II

Ryotaro Sato (2003)

Colloquium Mathematicae

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Previously we obtained stochastic and pointwise ergodic theorems for a continuous d-parameter additive process F in L₁((Ω,Σ,μ);X), where X is a reflexive Banach space, under the condition that F is bounded. In this paper we improve the previous results by considering the weaker condition that the function $W (\cdot) = e s s s u p {| | F (I) (\cdot) | | : I \subset [0, 1)}^{d}$ is integrable on Ω.

On ergodicity for operators with bounded resolvent in Banach spaces

Kirsti Mattila (2011)

Studia Mathematica

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We prove results on ergodicity, i.e. on the property that the space is a direct sum of the kernel of an operator and the closure of its range, for closed linear operators A such that ${| | α (α - A)}^{- 1} | |$ is uniformly bounded for all α > 0. We consider operators on Banach spaces which have the property that the space is complemented in its second dual space by a projection P. Results on ergodicity are obtained under a norm condition ||I - 2P|| ||I - Q|| < 2 where Q is a projection depending on the...

Generalizations of Cesàro means and poles of the resolvent

Laura Burlando (2004)

Studia Mathematica

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An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that $n^{- 2} T ⁿ$ converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and $n^{- 3} \sum_{k = 0}^{n - 1} A^{k}$ converges uniformly, the sum of the range of 1-A and the kernel of (1-A)²...

Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in $L^{p}$

Christophe Cuny (2011)

Colloquium Mathematicae

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We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^{p} (X, Σ, μ)$ , p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

Non-Typical Points for β-Shifts

David Färm, Tomas Persson (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study sets of non-typical points under the map $f_{β} \mapsto β x$ mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.

Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko, Anton V. Solomko (2009)

Colloquium Mathematicae

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We construct a rank-one infinite measure preserving flow ${(T_{r})}_{r \in ℝ}$ such that for each p > 0, the “diagonal” flow ${(T_{r} \times \dots \times T_{r})}_{r \in ℝ} (p t i m e s)$ on the product space is ergodic.

Ergodic averages with deterministic weights

Fabien Durand, Dominique Schneider (2002)

Annales de l’institut Fourier

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We study the convergence of the ergodic averages $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ where ${(θ (k))}_{k \in ℕ}$ is a bounded sequence and ${(u_{k})}_{k \in ℕ}$ a strictly increasing sequence of integers such that ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ for some $δ < 1$ . Moreover we give explicit such sequences $θ$ and $u$ and we investigate in particular the case where $θ$ is a $q$ -multiplicative sequence.

On ergodic properties of convolution operators associated with compact quantum groups

Uwe Franz, Adam Skalski (2008)

Colloquium Mathematicae

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Recent results of M. Junge and Q. Xu on the ergodic properties of the averages of kernels in noncommutative $L^{p}$ -spaces are applied to the analysis of almost uniform convergence of operators induced by convolutions on compact quantum groups.

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M ₙ^{α} = {(A ₙ^{α})}^{- 1} \sum_{k = 0}^{n} A_{n - k}^{α - 1} T^{k}$ . For α = 1, we find $M ₙ ¹ = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

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We call a sequence $(T_{n})$ of measure preserving transformations strongly mixing if $P (T_{n}^{- 1} A \cap B)$ tends to $P (A) P (B)$ for arbitrary measurable $A$ , $B$ . We investigate whether one can pass to a suitable subsequence $(T_{n_{k}})$ such that $\frac{1}{K} \sum_{k = 1}^{K} f (T_{n_{k}}) ⟶ \int f d P$ almost surely for all (or “many”) integrable $f$ .

Mean lower bounds for Markov operators

Eduard Emel&#039;yanov, Manfred Wolff (2004)

Annales Polonici Mathematici

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Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $(n^{- 1} \sum_{k = 0}^{n - 1} T^{k}) ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $l i m_{n \to \infty} | | (h - n^{- 1} \sum_{k = 0}^{n - 1} T^{k} f) ₊ | | = 0$ for every density f. Analogous results for strongly continuous semigroups are given.

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...

Ergodic averages with generalized weights

Doğan Çömez, Semyon N. Litvinov (2006)

Studia Mathematica

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Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are...