## Abstract

Let { h_{n}} be a sequence in R^{d} tending to infinity and let {Thn} be the corresponding sequence of shift operators given by (Thnf)(x)=f(x-hn) for x∈ R^{d}. We prove that {Thn} converges weakly to the zero operator as n→ ∞ on a separable rearrangement-invariant Banach function space X(R^{d}) if and only if its fundamental function φ_{X} satisfies φ_{X}(t) / t→ 0 as t→ ∞. On the other hand, we show that {Thn} does not converge weakly to the zero operator as n→ ∞ on all Marcinkiewicz endpoint spaces M_{φ}(R^{d}) and on all non-separable Orlicz spaces L^{Φ}(R^{d}). Finally, we prove that if { h_{n}} is an arithmetic progression: h_{n}= nh, n∈ N with an arbitrary h∈ R^{d}\ { 0 } , then { T_{nh}} does not converge weakly to the zero operator on any non-separable rearrangement-invariant Banach function space X(R^{d}) as n→ ∞.

Original language | English |
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Pages (from-to) | 91-124 |

Number of pages | 34 |

Journal | Revista Matematica Complutense |

Volume | 36 |

Issue number | 1 |

Early online date | 22 Mar 2022 |

DOIs | |

Publication status | Published - Jan 2023 |

## Keywords

- Fundamental function
- Limit operator
- Marcinkiewicz endpoint space
- Non-separable Orlicz space
- Rearrangement-invariant Banach function space
- Shift operator
- Weak convergence to zero