Rotundity and monotonicity properties of selected Cesàro function spaces. (English) Zbl 1398.46013
Summary: We study rotundity, strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some classes of Cesàro function spaces. We present full criteria of these properties in the Cesàro-Orlicz function spaces \(\mathrm{Ces}_{\phi}\) (under some mild assumptions on the Orlicz function \(\phi \)). Next, we prove a characterization of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in the Cesàro-Lorentz function spaces \(C\Lambda_{\phi }\). We also show that the space \(C\Lambda_{\phi}\) is never rotund. Finally, we will prove that Cesàro-Marcinkiewicz function space \(CM_{\phi}^{(\ast)}\) is neither strictly monotone nor order continuous for any quasi-concave function \(\phi\).
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46B04 | Isometric theory of Banach spaces |
| 46B42 | Banach lattices |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
Cesàro function space; Cesàro-Orlicz function space; Cesàro-Lorentz function space; Cesàro-Marcinkiewicz function space; rotundity; monotonicity properties; order continuityReferences:
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