Karlovych, OleksiyShargorodsky, Eugene2023-04-282024-01-022023-011139-1138PURE: 56423613PURE UUID: 49b5c778-cf84-4e0f-a810-a5058d847e8aScopus: 85126854346WOS: 000771853200001http://hdl.handle.net/10362/152251Publisher Copyright: © 2022, Universidad Complutense de Madrid.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→ ∞.34398264engFundamental functionLimit operatorMarcinkiewicz endpoint spaceNon-separable Orlicz spaceRearrangement-invariant Banach function spaceShift operatorWeak convergence to zeroGeneral Mathematicsjournal article10.1007/s13163-022-00423-4https://www.scopus.com/pages/publications/85126854346