Discrete Riesz transforms on rearrangement-invariant Banach sequence spaces and maximally noncompact operators

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 $Y(\mathbb{R}^d)$.