Let T and V be two Hilbert space contractions and let X be a linear bounded operator. It was proved by C. Foiaş and J. P. Williams that in certain cases the operator block matrix R(X;T,V) (equation (1.1) below) is similar to a contraction if and only if the commutator equation X = TZ-ZV has a bounded solution Z. We characterize here the similarity to contractions of some operator matrices R(X;T,V) in terms of growth conditions or of perturbations of R(0;T,V) = T ⊕ V.

Suppose A is a sectorial operator on a Banach space X, which admits an H∞-calculus. We study conditions on a multiplicative perturbation B of A which ensure that B also has an H∞-calculus. We identify a class of bounded operators T : X→X, which we call strongly triangular, such that if B = (1 + T) A is sectorial then it also has an H∞-calculus. In the case X is a Hilbert space an operator is strongly triangular if and only if ∑ Sn(T)/n <∞ where (Sn(T))n=1∞ are the singular values of T.

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

We answer a question of H. Furstenberg on the pointwise convergence of the averages $1/N{\sum }_{n=1}^{N}Uⁿ\left(f·Rⁿ\left(g\right)\right)$, where U and R are positive operators. We also study the pointwise convergence of the averages $1/N{\sum }_{n=1}^{N}f\left(Sⁿx\right)g\left(Rⁿx\right)$ when T and S are measure preserving transformations.

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...