## Abstract

Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X^{′}(R). The algebra C_{X}(R˙) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on R˙ = R∪ { ∞} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra A_{X}_{(}_{R}_{)} generated by all multiplication operators aI by continuous functions a∈ C(R˙) and by all Fourier convolution operators W(b) with symbols b∈ C_{X}(R˙). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra A_{X}_{(}_{R}_{)} does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces L^{p}^{,}^{1}(R) with 1 < p< ∞.

Original language | English |
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Article number | 16 |

Journal | Integral Equations And Operator Theory |

Volume | 93 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2021 |

## Keywords

- Algebra of convolution type operators
- Continuous Fourier multiplier
- Hardy-Littlewood maximal operator
- Lorentz space
- Rank one operator
- Separable Banach function space