### 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 language | English (US) |
---|---|

Pages (from-to) | 229-253 |

Number of pages | 25 |

Journal | Acta Mathematica Hungarica |

Volume | 157 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)