### On the Limit of Weighted Operator Averages.

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We give a counterexample showing that $\overline{(I-T*){L}_{\infty}}\cap {L}_{\infty}^{+}=0$ 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.

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 ${n}^{-1}{\sum}_{i=0}^{n-1}f\circ {\tau}^{i}\left(x\right)$ 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,or1<r<\infty and1\le s\le \infty ,orr=s=\infty $. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified...

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...

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.

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 ≠ ∞.

Let $T=T\left(u\right):u\in {\mathbb{R}}_{d}^{+}$ be a strongly continuous d-dimensional semigroup of linear contractions on ${L}_{1}((\Omega ,\Sigma ,\mu );X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since ${L}_{1}((\Omega ,\Sigma ,\mu );X)*={L}_{\infty}((\Omega ,\Sigma ,\mu );X*)$, the adjoint semigroup $T*=T*\left(u\right):u\in {\mathbb{R}}_{d}^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on ${L}_{\infty}((\Omega ,\Sigma ,\mu );X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u\in {\mathbb{R}}_{d}^{+}$, has a contraction majorant P(u) defined on ${L}_{1}((\Omega ,\Sigma ,\mu );\mathbb{R})$, that is, P(u) is a positive linear contraction on ${L}_{1}((\Omega ,\Sigma ,\mu );\mathbb{R})$ such that $\Vert T\left(u\right)f\left(\omega \right)\Vert \le P\left(u\right)\Vert f\left(\xb7\right)\Vert \left(\omega \right)$ almost everywhere...

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