On the weak convergence of shift operators to zero on rearrangement-invariant spaces

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→ ∞.