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 $W(·) = ess sup{||F(I)(·)||: I ⊂ [0, 1)^{d}}$ is integrable on Ω.
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Let τ be a null preserving point transformation on a finite measure space. Assuming τ is invertible, P. Ortega Salvador has recently obtained sufficient conditions for the almost everywhere convergence of the ergodic averages in $L_{pq}$ with 1 < p < ∞, 1 < q < ∞. In this paper we obtain necessary and sufficient conditions for the almost everywhere convergence, without assuming that τ is invertible and only assuming that p ≠ ∞.
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Let $T = {T(u): u ∈ ℝ_d^{+}}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((Ω,Σ,μ);X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((Ω,Σ,μ);X)* = L_∞((Ω,Σ,μ);X*)$, the adjoint semigroup $T* = {T*(u): u ∈ ℝ_d^{+}}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_∞((Ω,Σ,μ);X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u ∈ ℝ_d^{+}$, has a contraction majorant P(u) defined on $L_1((Ω,Σ,μ);ℝ)$, that is, P(u) is a positive linear contraction on $L_1((Ω,Σ,μ);ℝ)$ such that $‖T(u)f(ω)‖ ≤ P(u)‖f(·)‖(ω)$ almost everywhere on Ω for every $⨍ ∈ L_1((Ω,Σ,μ);X)$, we prove that the local ergodic theorem holds for T*.
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We give a counterexample showing that $\overline{(I-T*)L_{∞}} ∩ L^{+}_{∞} = {0}$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.
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Let X be a Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let L be a Banach space of X-valued strongly measurable functions on (Ω,Σ,μ). We consider a strongly continuous d-dimensional semigroup $T = {T(u): u = (u₁,...,u_{d})$, $u_{i} > 0$, 1 ≤ i ≤ d} of linear contractions on L. We assume that each T(u) has, in a sense, a contraction majorant and that the strong limit $T(0) = strong-lim_{u→0} T(u)$ exists. Then we prove, under some suitable norm conditions on the Banach space L, that a differentiation theorem holds for d-dimensional bounded processes in L which are additive with respect to the semigroup T. This generalizes a differentiation theorem obtained previously by the author under the assumption that L is an X-valued $L_{p}$-space, with 1 ≤ p < ∞.
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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 $n^{-1} ∑^{n-1}_{i=0} f∘τ^{i}(x)$ 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 ${(r,s) : r=s=1, or 1 < r < ∞ and 1 ≤ s ≤ ∞, or r = s = ∞}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified
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Let T be an endomorphism of a probability measure space (Ω,𝓐,μ), and f be a real-valued measurable function on Ω. We consider the cohomology equation f = h ∘ T - h. Conditions for the existence of real-valued measurable solutions h in some function spaces are deduced. The results obtained generalize and improve a recent result of Alonso, Hong and Obaya.
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Let X be a reflexive Banach space and (Ω,𝓐,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,𝓐,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $lim_{n→∞} fₙ = f$ in measure for some f ∈ M(μ;X), then also $lim_{n→∞} Tfₙ = Tf$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying f = (T-I)h.
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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 for a d-parameter bounded additive process F in $L_{1}$((Ω,Σ,μ);X) with respect to the semigroup T.
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Let $(L,||·||_{L})$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T = {T(u): u = (u₁,...,u_{d}), $u_{i} > 0$, 1 ≤ i ≤ d}$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.
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Let L be a Banach lattice of real-valued measurable functions on a σ-finite measure space and T={$T_t$: t < 0} be a strongly continuous semigroup of positive linear operators on the Banach lattice L. Under some suitable norm conditions on L we prove a general differentiation theorem for superadditive processes in L with respect to the semigroup T.
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We discuss implication relations for boundedness and growth orders of Cesàro means and Abel means of discrete semigroups and continuous semigroups of linear operators. Counterexamples are constructed to show that implication relations between two Cesàro means of different orders or between Cesàro means and Abel means are in general strict, except when the space has dimension one or two.