Displaying similar documents to “Generalizations of Cesàro means and poles of the resolvent”

Norm convergence of some power series of operators in L p with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of L p ( μ ) , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that s u p n | | U | | < . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like n 1 ( U f ) / n 1 - α , 0 ≤ α < 1, in terms of | | f + + U n - 1 f | | p , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

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 | | T | | / n 1 - γ = . 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 1 - γ = 0 . In this note we construct an example of a nonpositive L¹ operator with the...

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 α = ( A α ) - 1 k = 0 n A n - k α - 1 T k . For α = 1, we find M ¹ = ( n + 1 ) - 1 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.

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure m G . We consider the map τ φ defined on X × G by τ φ : ( x , g ) ( τ x , φ ( x ) g ) and the cocycle ( φ ) n generated by φ. Using a characterization of the ergodic invariant measures for τ φ , we give the form of the ergodic decomposition of μ ( d x ) m G ( d g ) or more generally of the τ φ -invariant measures μ χ ( d x ) χ ( g ) m G ( d g ) , where μ χ ( d x ) is χ∘φ-conformal for an exponential χ on G.

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with s u p | | T / n w | | < for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) n - 1 k = 0 n - 1 T k converges uniformly; (ii) c l ( I - T ) X = z X : l i m n k = 1 n T k z / k e x i s t s .

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 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 | | T | | / n 1 - γ = . 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 s u c h t h a t lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 . A class of numerical sequences αₙ, intimately...

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages A k f may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence A m k f of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are A k f ( x ) = 1 / ( 2 k ) j = 4 k + 1 4 k + 2 k f ( T j x ) , then the subsequence A k ² f will not be pointwise good even on L , but the subsequence A 2 k f will be pointwise good on L¹. Understanding when the hyperexponential...

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on L p ( ν ) , T , . . . , T k , n ̅ = ( n , . . . , n k ) k and α ̅ = ( α , . . . , α k ) with 0 < α j 1 , we define the ergodic Cesàro-α̅ averages n ̅ , α ̅ f = 1 / ( j = 1 k A n j α j ) i k = 0 n k i = 0 n j = 1 k A n j - i j α j - 1 T k i k T i f . For these averages we prove the almost everywhere convergence on X and the convergence in the L p ( ν ) norm, when n , . . . , n k independently, for all f L p ( d ν ) with p > 1/α⁎ where α = m i n 1 j k α j . In the limit case p = 1/α⁎, we prove that the averages n ̅ , α ̅ f converge almost everywhere on X for all f in the Orlicz-Lorentz space Λ ( 1 / α , φ m - 1 ) with φ ( t ) = t ( 1 + l o g t ) m . To obtain the result in the limit case we need...

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

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Let U , . . . , U d be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of d and the associated collection of rectangular parallelepipeds in d with sides parallel to the axes and dimensions of the form n × × n d with ( n , . . . , n d ) Γ . The associated multiparameter geometric and ergodic maximal operators M and M Γ are defined respectively on L ¹ ( d ) and L¹(Ω) by M g ( x ) = s u p x R 1 / | R | R | g ( y ) | d y and M Γ f ( ω ) = s u p ( n , . . . , n d ) Γ 1 / n n d j = 0 n - 1 j d = 0 n d - 1 | f ( U j U d j d ω ) | . Given a Young function Φ, it is shown that M satisfies the weak type estimate ...

The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen, Christophe Cuny, Michael Lin (2010)

Studia Mathematica

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Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform l i m n k = 1 n ( T k x ) / k . We prove that weak and strong convergence are equivalent, and in a reflexive space also s u p n | | k = 1 n ( T k x ) / k | | < is equivalent to the convergence. We also show that - k = 1 ( T k ) / k (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ( I - T ) | ( I - T ) X ¯ r .

Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let f V r ( ) r ( ) , where, for 1 ≤ r < ∞, V r ( ) (resp., r ( ) ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

On the convergence to 0 of mₙξmod 1

Bassam Fayad, Jean-Paul Thouvenot (2014)

Acta Arithmetica

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We show that for any irrational number α and a sequence m l l of integers such that l i m l | | | m l α | | | = 0 , there exists a continuous measure μ on the circle such that l i m l | | | m l θ | | | d μ ( θ ) = 0 . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence m l l of integers such that | | | m l α | | | 0 and such that m l θ [ 1 ] is dense on the circle if and only if θ ∉ ℚα + ℚ.

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval E may be defined as a uniform integrability property that involves both the integrand f n and the corresponding primitive F n . The pointwise convergence of the integrands f n to some f and the equiintegrability of the functions f n together imply that f is also integrable with primitive F and that the primitives F n converge uniformly to F . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Distortion bounds for C 2 + η unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

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We obtain estimates for derivative and cross-ratio distortion for C 2 + η (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that g T : T is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

Precompactness in the uniform ergodic theory

Yu. Lyubich, J. Zemánek (1994)

Studia Mathematica

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We characterize the Banach space operators T whose arithmetic means n - 1 ( I + T + . . . + T n - 1 ) n 1 form a precompact set in the operator norm topology. This occurs if and only if the sequence n - 1 T n n 1 is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.

Existence and upper semicontinuity of uniform attractors in H ¹ ( N ) for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors ε in the space H ¹ ( N ) for the nonautonomous nonclassical diffusion equation u t - ε Δ u t - Δ u + f ( x , u ) + λ u = g ( x , t ) , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ε ε [ 0 , 1 ] at ε = 0 is also studied.

Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the d case and obtain an explicit construction of a generic point of the d shift transformation of the set 0 , 1 , . . . , b - 1 d .

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...

On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

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Two problems posed by Choquet and Foias are solved: (i) Let T be a positive linear operator on the space C ( X ) of continuous real-valued functions on a compact Hausdorff space X . It is shown that if n - 1 r = 0 n - 1 T r 1 converges pointwise to a continuous limit, then the convergence is uniform on X . (ii) An example is given of a Choquet simplex K and a positive linear operator T on the space A ( K ) of continuous affine real-valued functions on K , such that inf { ( T n 1 ) ( x ) : n } &lt; 1 for each...

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) k = 1 a k s i n k x be given with nonnegative coefficients a k . If a k is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that k a k 0 as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) k = 1 l = 1 c k l s i n k x s i n l y , even with complex coefficients c k l . We also...

Bounded evaluation operators from H p into q

Martin Smith (2007)

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by T z , p ( f ) = ( 1 - | z | ² ) 1 / p f ( z ) . Necessary and sufficient conditions on zₙ are given such that T z , p maps the Hardy space H p boundedly into the sequence space q . A corresponding result for Bergman spaces is also stated.

Some remarks on universality properties of / c

Mikołaj Krupski, Witold Marciszewski (2012)

Colloquium Mathematicae

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We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into / c . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into / c , but fails to embed...

A uniform dichotomy for generic SL ( 2 , ) cocycles over a minimal base

Artur Avila, Jairo Bochi (2007)

Bulletin de la Société Mathématique de France

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We consider continuous SL ( 2 , ) -cocycles over a minimal homeomorphism of a compact set K of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....