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 {T-hn} be the corresponding sequence of shift operators given by (T-hn f)(x) = f ( x - h(n)) for x is an element of R-d. We prove that {T-hn} converges weakly to the zero operator as n -> infinity on a separable rearrangement-invariant Banach function space X(R-d) if and only if its fundamental function phi(X) satisfies phi(X) (t)/t -> 0 as t -> infinity. On the other hand, we show that {T-hn} does not converge weakly to the zero operator as n -> infinity on all Marcinkiewicz endpoint spaces M-phi(R-d) and on all non-separable Orlicz spaces L-Phi(R-d). Finally, we prove that if {h(n)} is an arithmetic progression: h(n) = nh, n is an element of N with an arbitrary h is an element of 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 -> infinity.