Noncommutative almost uniform Wiener-Wintner ergodic theorem
Almost uniform version of noncommutative Wiener-Wintner ergodic theorem and its extension to Besicovitch weights are proved.
math.FA↗
arXiv subjects
Vladimir Chilin
Publications and source records attributed to Vladimir Chilin.
On individual ergodic theorems for semifinite von Neumann algebras
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.
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Individual ergodic theorems for semifinite von Neumann algebras
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.
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Almost uniform convergence in Wiener-Wintner ergodic theorem
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.
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Skew-Hermitian operators in real Banach spaces of self-adjoint compact operators
Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a symmetric sequence space. If $\{μ(n,x)\}$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{μ(n,x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{μ(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$. We show that in the case when $\mathcal C_E$ is a separable or perfect Banach symmetric ideal, $\mathcal C_E \neq \mathcal C_{l_2}$, for any skew-Hermitian operator $H\colon\mathcal C_E^h \to \mathcal C_E^h$ there exists self-adjoint bounded linear operator $a$ in $\mathcal H$ such that $H(x)=i(xa - ax)$ for all $x\in\mathcal C_E^h$.
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Individual ergodic theorems for infinite measure
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.
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Ergodic theorems in Banach ideals of compact operators
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.
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Noncommutative weighted individual ergodic theorems with continuous time
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.
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Local ergodic theorems in symmetric spaces of measurable operators
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.
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Almost uniform and strong convergences in ergodic theorems for symmetric spaces
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 $X$.
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Isomorphic classification of $\ast$-algebras of log-integrable functions
Using the notion of passport of a normed Boolean algebra, necessary and sufficient conditions for a $\ast$-isomorphism of $\ast$-algebras of log-integrable measurable functions are found.
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Uniform convergence in the individual ergodic theorem for symmetric sequence spaces
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 $\| \frac 1n \sum_{k=0}^{n-1}T^k(x) - \widehat x \|_\infty \to 0$.
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The validity space of Dunford-Schwartz ergodic theorem for infinite measure
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 $λ> 0$.
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Validity space of Dunford-Schwartz pointwise ergodic theorem
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 $λ>0$.
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Isometries of perfect norm ideals of compact operators
It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $\mathcal C_E\neq \mathcal C_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $\mathcal H$ there exist unitary operators $u$ and $v$ on $\mathcal H$ such that $V(x)=uxv$ or $V(x) = ux^tv$ for all $x\in \mathcal C_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $\mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $\mathcal C_E \neq \mathcal C_2$ is a linear isometry on $\mathcal C_E$.
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Pointwise ergodic theorems in symmetric spaces of measurable functions
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.
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On pointwise ergodic theorems for infinite measure
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.
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Lie derivations on the algebras of locally measurable operators
We prove that every Lie derivation on a solid $\star-$subalgebras of locally measurable operators it is equal to a sum of the associative derivation and the center-valued trace.
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