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 (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 T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p&gt;1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe....

We distinguish a class of unbounded operators in ${}^r$, r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in ${}^r$-spaces are applied.

In 1967, Ross and Stromberg published a theorem about pointwise limits of orbital integrals for the left action of a locally compact group G on (G,ρ), where ρ is the right Haar measure. We study the same kind of problem, but more generally for left actions of G on any measure space (X,μ), which leave the σ-finite measure μ relatively invariant, in the sense that sμ = Δ(s)μ for every s ∈ G, where Δ is the modular function of G. As a consequence, we also obtain a generalization of a theorem of Civin...

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 $x\in X:su{p}_n\left|\right|{\sum }_{k=1}^n{T}^{k}x\left|\right|<\infty$ = (I-T)X$.$We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_t:t\ge 0$ with generator A satisfies $AX=x\in X:su{p}_{s>0}\left|\right|{\int }_0^s{T}_{t}xdt\left|\right|<\infty$. 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...

We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators satisfying the Ritt resolvent condition: compare Theorem 5 and Theorem 9 below.

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, $\left|\right|Tⁿ\left|\right|\sim n^{1/4}$). 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 $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty .IntheclassofpositiveoperatorstheseexamplesarethebestpossibleinthesensethatforeverysuchoperatorTthereexistsa\gamma ₀>0suchthat$lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$.$A class of numerical sequences αₙ, intimately related to the...

Firstly, we give extensions of results of Gelfand, Esterle and Katznelson--Tzafriri on power-bounded operators. Secondly, some results and questions relating to power-bounded elements in the unitization of a commutative radical Banach algebra are discussed.

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

Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀:=x\in X:li{m}_{n\to \infty }\left|\right|Tⁿx\left|\right|=0$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness ${\chi }_{\left|\right|·\left|\right|₁}\left(A\right)<1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove that every...