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On continuity at zero of the maximal operator for a semifinite measure

100%
Litvinov S.
Colloquium Mathematicum
|
2014
|
tom 135
|
nr 1
79-84
EN
For a sequence of linear maps defined on a Banach space with values in the space of measurable functions on a semifinite measure space, we examine the behavior of its maximal operator at zero.
2
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On individual subsequential ergodic theorem in von Neumann algebras

63%
Litvinov S., Mukhamedov F.
Studia Mathematica
|
2001
|
tom 145
|
nr 1
55-62
EN
We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting.
3
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Banach principle in the space of τ-measurable operators

63%
Goldstein M., Litvinov S.
Studia Mathematica
|
2000
|
tom 143
|
nr 1
33-41
EN
We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.
4
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Ergodic theorems in fully symmetric spaces of τ-measurable operators

63%
Chilin V., Litvinov S.
Studia Mathematica
|
2015
|
tom 228
|
nr 2
177-195
EN
Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$-spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$-spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with the Fatou property and nontrivial Boyd indices, in particular, to noncommutative Lorentz spaces $L_{p,q}$. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.
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