Date of Award
Doctor of Philosophy
Orlicz-Lorentz function and sequence spaces, which appeared asinterpolation spaces between Lorentz spaces and L infinity via Calderon's method ofinterpolation, are an important class of Kothe spaces. The recent discovery of theirexplicit Kothe duals has provided more opportunities to study geometrical properties of Orlicz-Lorentz spaces.First, we present basic properties of the Kothe duals of Orlicz-Lorentz spaces.We provideexplicit formulas for the Orlicz norm in these spaces. In addition, we characterizeseparability of these spaces. Consequently, we obtain a characterization of the spaces containing isomorphic and isometric copies of c0 and l infinity.Second, we consider the M-ideal properties in Orlicz-Lorentz function andsequence spaces. The main results in this part are explicit formulas for the norm ofbounded linear functionals on these spaces equipped with the Luxemburg norm orthe Orlicz norm. Subsequently, the order-continuous subspaces are M-ideals if thespaces are equipped with the Luxemburg norm, while this is not the case with theOrlicz norm. As an application, we provide a characterization of norm-attainingfunctionals on Orlicz-Lorentz function and sequence spaces. We also show theexistence of unique norm-preserving extensions of integral functionals for thefunction space. In addition, M-embeddedness of Orlicz-Lorentz spaces is studied viathe geometric version of the M-ideal property described by the intersection of balls.Third, we study the u-ideal properties, which are more general than the M-idealproperties, of Orlicz-Lorentz spaces. We provide a characterization of strict u-idealsin biduals for Orlicz-Lorentz spaces with the existence of an isomorphic copy of l1through the Matuszewska-Orilcz indices.Finally, we provide a characterization of Orlicz-Lorentz function and sequencespaces with the local diameter two property, the diameter two property, and theRadon-Nikodym property in terms of a certain growth condition. From our M-embeddednessresult, we can obtain a sufficient condition for the strong diameter two property ofOrlicz-Lorentz function and sequence spaces. In addition, we provide a sufficientcondition for Orlicz-Lorentz spaces having the weak* strong diameter two property.
dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Tag, Hyung-Joon, "Some Geometrical Properties of Orlicz-Lorentz Spaces and their Kothe Duals" (2017). Electronic Theses and Dissertations. 1636.