## Abstract

We say that an operator between Banach spaces is maximally noncompact if its

operator norm coincides with its Hausdorff measure of noncompactness. We prove

that a translation-invariant operator acting from a translation-invariant

Banach sequence space $X(\mathbb{Z}^d)$ to a translation-invariant Banach

sequence space $Y(\mathbb{Z}^d)$ is maximally noncompact whenever the target

space $Y(\mathbb{Z}^d)$ satisfies mild additional conditions. As a consequence, we show

that the discrete Riesz transforms $R_j$, $j=1,\dots,d$ on

rearrangement-invariant Banach sequence spaces with non-trivial Boyd

indices are maximally noncompact. We also observe that the same results

are valid for translation-invariant operators between translation-invariant

Banach function spaces $X(\mathbb{R}^d)$ and $Y(\mathbb{R}^d)$.

operator norm coincides with its Hausdorff measure of noncompactness. We prove

that a translation-invariant operator acting from a translation-invariant

Banach sequence space $X(\mathbb{Z}^d)$ to a translation-invariant Banach

sequence space $Y(\mathbb{Z}^d)$ is maximally noncompact whenever the target

space $Y(\mathbb{Z}^d)$ satisfies mild additional conditions. As a consequence, we show

that the discrete Riesz transforms $R_j$, $j=1,\dots,d$ on

rearrangement-invariant Banach sequence spaces with non-trivial Boyd

indices are maximally noncompact. We also observe that the same results

are valid for translation-invariant operators between translation-invariant

Banach function spaces $X(\mathbb{R}^d)$ and $Y(\mathbb{R}^d)$.

Original language | English |
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Pages (from-to) | 195-210 |

Journal | Pure and Applied Functional Analysis |

Volume | 9 |

Issue number | 1 |

Publication status | Published - 7 Mar 2024 |

## Keywords

- Discrete Riesz transform
- translation-invariant operator
- Hausdorff measure of noncompactness
- essential norm
- translation-invariant space, rearrangement-invariant space
- Boyd indices