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→ ∞.