TY - JOUR
T1 - Almost uniform and strong convergences in ergodic theorems for symmetric spaces
AU - Chilin, V.
AU - Litvinov, S.
N1 - Publisher Copyright:
© 2018, Akadémiai Kiadó, Budapest, Hungary.
PY - 2019/2/1
Y1 - 2019/2/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s10474-018-0872-1
DO - 10.1007/s10474-018-0872-1
M3 - Article
AN - SCOPUS:85053791001
SN - 0236-5294
VL - 157
SP - 229
EP - 253
JO - Acta Mathematica Hungarica
JF - Acta Mathematica Hungarica
IS - 1
ER -