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

Abstract

Let { hn} be a sequence in Rd tending to infinity and let {Thn} be the corresponding sequence of shift operators given by (Thnf)(x)=f(x-hn) for x∈ Rd. We prove that {Thn} converges weakly to the zero operator as n→ ∞ on a separable rearrangement-invariant Banach function space X(Rd) if and only if its fundamental function φX satisfies φX(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 Mφ(Rd) and on all non-separable Orlicz spaces LΦ(Rd). Finally, we prove that if { hn} is an arithmetic progression: hn= nh, n∈ N with an arbitrary h∈ Rd\ { 0 } , then { Tnh} does not converge weakly to the zero operator on any non-separable rearrangement-invariant Banach function space X(Rd) as n→ ∞.