Abstract
We describe the spectral properties of the dilation operator \( Tx(t)=x(t/2) \), with \( t>0 \), in separable rearrangement invariant spaces of fundamental type. We show that these properties are determined by the values of the dilation indices of the spaces and apply the results to Orlicz spaces.
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Notes
In [4], we use a somewhat different definition of the norm in \( \Lambda_{p}(\psi) \); namely, \( \|x\|_{\Lambda_{p,\psi}}:=\big{(}\!\int\nolimits_{0}^{\infty}x^{*}(t)^{p}\psi(t)^{p}\,\frac{dt}{t}\big{)}^{1/p} \).
The first example of an r.i. space of nonfundamental type was constructed by Shimogaki in [16].
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Funding
The work was carried out in the framework of the Development Program of the Scientific and Educational Mathematical Center of the Volga Federal District (Agreement 075–02–2020–1488/1).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 1, pp. 3–16. https://doi.org/10.33048/smzh.2023.64.101
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Astashkin, S.V. Spectral Properties of the Dilation Operator in Rearrangement Invariant Spaces of Fundamental Type. Sib Math J 64, 1–12 (2023). https://doi.org/10.1134/S0037446623010019
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DOI: https://doi.org/10.1134/S0037446623010019