This article gives an affirmative solution to the problem whether the ergodic Cesáro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,τ)$, $1\leq p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon, published in 1977, where bilaterally almost uniform convergence of these averages was established for $p=1$.
We show that if a $σ-$finite infinite measure space $(Ω,μ)$ is quasi-non-atomic, then the Dunford-Schwartz pointwise ergodic theorem holds for $f\in \mathcal L^1(Ω)+\mathcal L^{\infty}(Ω)$ if and only if $μ\{f\ge λ\}<\infty$ for all $λ>0$.
In this paper, we state as a conjecture a vector-valued Hopf-Dunford-Schwartz lemma and give a partial answer to it. As an application of this powerful result, we prove some Fe fferman-Stein inequalities in the setting of Dunkl analysis where the classical tools of real analysis cannot be applied.
We show that if $(Ω,μ)$ is an infinite measure space, the pointwise Dunford-Shwartz ergodic theorem holds for $f \in \mathcal L^1(Ω)+\mathcal L^\infty(Ω)$ if and only if $μ\{f>λ\}<\infty$ for all $λ> 0$.
An integral for a scalar function with respect to a multimeasure $N$ taking its values in a locally convex space is introduced. The definition is independent of the selections of $N$ and is related to a functional version of the Bartle-Dunford-Schwartz integral with respect to a vector measure presented by Lewis. Its properties are studied together with its application to Radon-Nikodym theorems in order to represent as an integrable derivative the ratio of two general multimeasures or two $d_H$-multimeasures; equivalent conditions are provided in both cases.
Let $X$ and $Y$ be Banach spaces and let $Ω$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(Ω,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(Ω),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(Ω,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:Σ\rightarrow \mathcal{L}(X,Y^{**})$, where $Σ$ denotes the $σ$-algebra of Borel subsets of $Ω$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, whi…
Extending a result by Chilin and Litvinov, we show by construction that given any $σ$-finite infinite measure space $(Ω,\mathcal{A}, μ)$ and a function $f\in L^1(Ω)+L^\infty(Ω)$ with $μ(\{|f|>\varepsilon\})=\infty$ for some $\varepsilon>0$, there exists a Dunford-Schwartz operator $T$ over $(Ω,\mathcal{A}, μ)$ such that $\frac{1}{N}\sum_{n=1}^N (T^nf)(x)$ fails to converge for almost every $x\inΩ$. In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $L^1(Ω)+L^\infty(Ω)$.
We investigate pointwise convergence of entangled ergodic averages of Dunford-Schwartz operators $T_0,T_1,\ldots, T_m$ on a Borel probability space. These averages take the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{α(m)}}A_{m-1}T^{n_{α(m-1)}}_{m-1}\ldots A_2T_2^{n_{α(2)}}A_1T_1^{n_{α(1)}} f, \] where $f\in L^p(X,μ)$ for some $1\leq p<\infty$, and $α:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ encodes the entanglement. We prove that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for a…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x)\to 0$ as $t\to\infty$ for every $x\in E$, where $μ_t(x)$ is the non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmet…
Given a $σ$-finite infinite measure space $(Ω,μ)$, it is shown that any Dunford-Schwartz operator $T:\,\mathcal L^1(Ω)\to\mathcal L^1(Ω)$ can be uniquely extended to the space $\mathcal L^1(Ω)+\mathcal L^\infty(Ω)$. This allows to find the largest subspace $\mathcal R_μ$ of $\mathcal L^1(Ω)+\mathcal L^\infty(Ω)$ such that the ergodic averages $\frac1n\sum\limits_{k=0}^{n-1}T^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\in\mathcal R_μ$ and every Dunford-Schwartz operator $T$. Utilizing this result, almost uniform convergence of the averages $\frac1n\sum\limits_{k=0}^{n-1}β_k…
Let $(Ω,μ)$ be a $σ$-finite measure space, and let $X\subset L^1(Ω)+L^\infty(Ω)$ be a fully symmetric space of measurable functions on $(Ω,μ)$. If $μ(Ω)=\infty$, necessary and sufficient conditions are given for almost uniform convergence in $X$ (in Egorov's sense) of Cesàro averages $M_n(T)(f)=\frac1n\sum_{k = 0}^{n-1}T^k(f)$ for all Dunford-Schwartz operators $T$ in $L^1(Ω)+ L^\infty(Ω)$ and any $f\in X$. Besides, it is proved that the averages $M_n(T)$ converge strongly in $X$ for each Dunford-Schwartz operator $T$ in $L^1(Ω)+L^\infty(Ω)$ if and only if $X$ has order continuous norm and $L^…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages.
It is proved that for any Dunford-Schwartz operator $T$ acting in the space $l_\infty$ and for each $x\in c_0 $ there exists an element $\widehat x \in c_0 $ such that $\| \frac 1n \sum_{k=0}^{n-1}T^k(x) - \widehat x \|_\infty \to 0$.
Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\l…
For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of ergodic averages in fully symmetric spaces of measurable functions with non-trivial Boyd indices is studied. In particular, it is shown that for such spaces Bourgain's Return Times theorem is valid.
We show that ergodic flows in noncommutative fully symmetric spaces (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also discussed.
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge bilaterally almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. In particular, these averages converge bilaterally almost uniformly in all noncommutative symmetric spaces with…
This paper is devoted to the study of various maximal ergodic theorems in noncommutative $L_p$-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on $L_p$ and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in theory of von Neumann algebras and in quantum probability.
We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$ \sum_{n=1}^N T_a^n A_{a-1}T_{a-1}^nA_{a-1}\cdot \ldots \cdot A_0 T_0^nf$$ for $f\in L^p(X,μ)$, $p\geq 1$. We also discuss examples and present a continuous version of the result.
This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $α$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by \begin{equation*} A_{n}x= \frac{1}{m(V^{n})} \int _{V^{n}}α_{g}x\,dm(g),\quad x\in L_{p}( \mathcal{M}),n\in \mathbb{N},1\leq p\leq \infty , \end{equation*} are of weak type $(1,1)$ and of strong type $(p,p)$ for …
We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.
Suppose that T_t is a symmetric diffusion semigroup on L^2(X) and consider its tensor product extension to the Bochner space L^p(X,B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of the semigroup's extension to L^p(X,B). As an application, we show that such continuations exhibit pointwise convergence.
