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1
Content available remote

The uniform zero-two law for positive operators in Banach lattices

100%
Lin M.
Studia Mathematica
|
1998
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tom 131
|
nr 2
149-153
EN
Let T be a positive power-bounded operator on a Banach lattice. We prove: (i) If $inf_n ||T^n(I-T)|| < 2$, then there is a k ≥ 1 such that $lim_{n→∞} ||T^n(I-T^k)|| = 0. (ii) $lim_{n→∞} ||T^n(I-T)|| = 0$ if (and only if) $inf_n ||T^n(I-T)|| < √3$.
2
Content available remote

Support overlapping $L_{1}$ contractions and exact non-singular transformations

100%
Lin M.
Colloquium Mathematicum
|
2000
|
tom 84/85
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nr 2
515-520
EN
Let T be a positive linear contraction of $L_{1}$ of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.
3
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Averages of unitary representations and weak mixing of random walks

64%
Lin M., Wittmann R.
Studia Mathematica
|
1995
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tom 114
|
nr 2
127-145
EN
Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; $U^n$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^{-1} ∑_{k=1}^n |⟨μ^{k}*f,g⟩| → 0$ for $g ∈ L_∞(G)$ and $f ∈ L_{1}(G)$ with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of $μ^n$ on $UCB_{l}(G)$
4
Content available remote

Contractive projections on the fixed point set of $L_∞$ contractions

64%
Lin M., Sine R.
Colloquium Mathematicum
|
1991
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tom 62
|
nr 1
91-96
5
Content available remote

Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations

64%
Kornfeld I., Lin M.
Studia Mathematica
|
2000
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tom 138
|
nr 3
225-240
EN
It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_1$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_1$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function $φ ∈ L_∞$ with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the $L_1$ topology)
6
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The one-sided ergodic Hilbert transform in Banach spaces

51%
Cohen G., Cuny C., Lin M.
Studia Mathematica
|
2010
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tom 196
|
nr 3
251-263
EN
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 $lim_{n} ∑_{k=1}^{n} (T^{k}x)/k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $sup_{n} ||∑_{k=1}^{n} (T^{k}x)/k|| < ∞$ 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 $(I-T)^{r}_{|\overline{(I-T)X}}$.
7
Content available remote

On the uniform ergodic theorem in Banach spaces that do not contain duals

51%
Fonf V., Lin M., Rubinov A.
Studia Mathematica
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1996
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tom 121
|
nr 1
67-85
EN
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.
8
Content available remote

Poisson's equation and characterizations of reflexivity of Banach spaces

51%
Fonf V., Lin M., Wojtaszczyk P.
Colloquium Mathematicum
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2011
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tom 124
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nr 2
225-235
EN
Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder's equality $x ∈ X: sup_{n} ||∑_{k = 1}^{n} T^{k}x|| < ∞} = (I-T)X$. We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup ${T_{t}: t ≥ 0}$ with generator A satisfies $AX = {x ∈ X: sup_{s>0} ||∫_{0}^{s} T_{t}xdt|| < ∞}$. The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson's equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities for the ranges express sufficient (and obviously necessary) conditions for solvability of Poisson's equation.
9
Content available remote

Consistency of the LSE in Linear regression with stationary noise

51%
Cohen G., Lin M., Tempelman A.
Colloquium Mathematicum
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2004
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tom 100
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nr 1
29-71
EN
We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_{p}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance). When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences. We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE. The results are applied to amplitude estimation in sums of harmonic signals with known frequencies.
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