King's College London

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