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Isomorphic Structure of Cesàro and Tandori Spaces
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 09 January 2019, pp. 501-532
- Print publication:
- June 2019
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- Article
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We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space
$\text{Ces}_{\infty }$ and its sequence counterpart 
$\text{ces}_{\infty }$ are isomorphic. This is rather surprising since 
$\text{Ces}_{\infty }$ (like Talagrand’s example) has no natural lattice predual. We prove that 
$\text{ces}_{\infty }$ is not isomorphic to 
$\ell _{\infty }$ nor is 
$\text{Ces}_{\infty }$ isomorphic to the Tandori space 
$\widetilde{L_{1}}$ with the norm 
$\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$, where 
$\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$. Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.
