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

Abstract

Let \(\{h_n\}\) be a sequence in \({\mathbb {R}}^d\) tending to infinity and let \(\{T_{h_n}\}\) be the corresponding sequence of shift operators given by \((T_{h_n}f)(x)=f(x-h_n)\) for \(x\in {\mathbb {R}}^d\). We prove that \(\{T_{h_n}\}\) converges weakly to the zero operator as \(n\rightarrow \infty \) on a separable rearrangement-invariant Banach function space \(X({\mathbb {R}}^d)\) if and only if its fundamental function \(\varphi _X\) satisfies \(\varphi _X(t)/t\rightarrow 0\) as \(t\rightarrow \infty \). On the other hand, we show that \(\{T_{h_n}\}\) does not converge weakly to the zero operator as \(n\rightarrow \infty \) on all Marcinkiewicz endpoint spaces \(M_\varphi ({\mathbb {R}}^d)\) and on all non-separable Orlicz spaces \(L^\varPhi ({\mathbb {R}}^d)\). Finally, we prove that if \(\{h_n\}\) is an arithmetic progression: \(h_n = nh\), \(n \in {\mathbb {N}}\) with an arbitrary \(h\in {\mathbb {R}}^d{\setminus }\{0\}\), then \(\{T_{nh}\}\) does not converge weakly to the zero operator on any non-separable rearrangement-invariant Banach function space \(X({\mathbb {R}}^d)\) as \(n\rightarrow \infty \).

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Acknowledgements

We would like to thank the anonymous referees for the helpful remarks.

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Correspondence to Oleksiy Karlovych.

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This work was supported by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within the scope of the project UIDB/00297/2020 (Centro de Matemática e Aplicações).

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Karlovych, O., Shargorodsky, E. On the weak convergence of shift operators to zero on rearrangement-invariant spaces. Rev Mat Complut (2022). https://doi.org/10.1007/s13163-022-00423-4

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  • DOI: https://doi.org/10.1007/s13163-022-00423-4

Keywords

  • Rearrangement-invariant Banach function space
  • Marcinkiewicz endpoint space
  • Non-separable Orlicz space
  • Shift operator
  • Weak convergence to zero
  • Fundamental function
  • Limit operator

Mathematics Subject Classification

  • 46E30