arXiv subjects
Publications and source records attributed to Semyon Litvinov.
We present a concise self-contained inversive geometry solution of the three-circle problem of Steiner of constructing a circle that intersects each of the three given circles at one of the three given angles.
Given $1\leq p<\infty$, we show that ergodic flows in the $L^p$-space over a $σ$-finite measure space generated by strongly continuous semigroups of Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly (in Egorov's sense). The corresponding local ergodic theorem is proved with identification of the limit. Then we extend these results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces.
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 prove that the ergodic Ces\' aro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,τ)$, $1<p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon \cite{ye}, where bilaterally almost uniform convergence of these averages was established for $p=1$.
A simple proof of Egorov's theorem for infinite measure is given
Given a semifinite von Neumann algebra $\mathcal M$ equipped with a faithful normal semifinite trace $τ$, we prove that the spaces $L^0(\mathcal M,τ)$ and $\mathcal R_τ$ are complete with respect to pointwise, almost uniform and bilaterally almost uniform, convergences in $L^0(\mathcal M,τ)$. Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space $L^1(\mathcal M,τ)$ can be extended to pointwise convergence of such nets in any fully symmetric space $E\subset\mathcal R_τ$, in particular, in any space $L^p(\mathcal M,τ)$, $1\leq p<\infty$. Some applications of these results in the noncommutative ergodic theory are discussed.
Almost uniform version of noncommutative Wiener-Wintner ergodic theorem and its extension to Besicovitch weights are proved.
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 symmetric spaces for which Dunford-Schwartz-type individual ergodic theorems hold is outlined.
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 spaces for which Dunford-Schwartz-type individual ergodic theorems hold is outlined. Also, almost uniform convergence in noncommutative Wiener-Wintner theorem is proved.
We extend almost everywhere convergence in Wiener-Wintner ergodic theorem for $σ$-finite measure to a generally stronger almost uniform convergence and present a larger, universal, space for which this convergence holds. We then extend this result to the case with Besicovitch weights.
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 $\frac1n\sum\limits_{k=0}^{n-1}β_kT^k(f)$ for every $f\in\mathcal R_μ$, any Dunford-Schwartz operator $T$ and any bounded Besicovitch sequence $\{β_k\}$ is established. Further, given a measure preserving transformation $τ:Ω\toΩ$, Assani's extension of Bourgain's Return Times theorem to $σ$-finite measure is employed to show that for each $f\in\mathcal R_μ$ there exists a set $Ω_f\subsetΩ$ such that $μ(Ω\setminusΩ_f)=0$ and the averages $\frac1n\sum\limits_{k=0}^{n-1}β_kf(τ^kω)$ converge for all $ω\inΩ_f$ and any bounded Besicovitch sequence $\{β_k\}$. Applications to fully symmetric subspaces $E\subset\mathcal R_μ$ are given.
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 $A_n(T)(x)=\frac1{n+1}\sum\limits_{k = 0}^n T^k(x)$ converge uniformly in $\mathcal C_E$ for any positive Dunford-Schwartz operator $T$ and $x\in\mathcal C_E$. Besides, if $x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Dunford-Schwartz operator $T$ such that the sequence $\{A_n(T)(x)\}$ does not converge uniformly. We also show that the averages $A_n(T)$ converge strongly in $(\mathcal C_E,\|\cdot\|_{\mathcal C_E})$ if and only if $E$ is separable and $E\neq l^1$, as sets.
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.
Local mean and individual (with respect to almost uniform convergence in Egorov's sense) ergodic theorems are established for actions of the semigroup $\mathbb R_+^d$ in symmetric spaces of measurable operators associated with a semifinite von Neumann algebra.
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 $L^1(Ω)$ is not contained in $X$.
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 $λ> 0$.
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 $λ>0$.
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.