Almost uniform and strong convergences in ergodic theorems for symmetric spaces

V. Chilin, Semyon Litvinov

Research output: Contribution to journalArticle

Abstract

Let (Ω , μ) be a σ-finite measure space, and let X⊂ L 1 (Ω) + L (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L 1 (Ω) + L (Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M n (T) converge strongly in X for each Dunford–Schwartz operator T in L 1 (Ω) + L (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

Original languageEnglish (US)
Pages (from-to)229-253
Number of pages25
JournalActa Mathematica Hungarica
Volume157
Issue number1
DOIs
StatePublished - Feb 1 2019

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Almost Convergence
Ergodic Theorem
Uniform convergence
Symmetric Spaces
Strong Convergence
Order Continuous Norm
Measure space
Measurable function
Operator
If and only if
Converge
Necessary Conditions
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "Let (Ω , μ) be a σ-finite measure space, and let X⊂ L 1 (Ω) + L ∞ (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Ces{\`a}ro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L 1 (Ω) + L ∞ (Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M n (T) converge strongly in X for each Dunford–Schwartz operator T in L 1 (Ω) + L ∞ (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.",
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Almost uniform and strong convergences in ergodic theorems for symmetric spaces. / Chilin, V.; Litvinov, Semyon.

In: Acta Mathematica Hungarica, Vol. 157, No. 1, 01.02.2019, p. 229-253.

Research output: Contribution to journalArticle

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