On some geometric properties of generalized Orlicz-Lorentz sequence spaces
The author investigates several important geometric properties of generalized Orlicz-Lorentz sequence spaces \(\lambda _{\varphi }\). The spaces \(\lambda _{\varphi }\) cover three classes of symmetric Banach sequence spaces: classical Orlicz-Lorentz sequence spaces, two-weighted Orlicz-Lorentz sequence spaces and symmetrizations of Musielak-Orlicz sequence spaces of Nakano type. First, the author discusses relationships between norm and modular convergence which are crucial tools in verifying geometric properties. The condition \(\delta _{2}^{\lambda }\) and strong condition \(\delta _{2}^{\lambda }\) (introduced by the author) are very useful at that point. The author also describes these conditions in special cases of spaces \(\lambda _{\varphi }.\) Next, the author characterizes the Kadec-Klee properties with respect to cordinatewise (uniform) convergence. The next section is devoted to monotonicity properties of the spaces \(\lambda _{\varphi }\). It is known that monotonicity properties (strict and uniform monotonicity) play ana\-lo\-gous roles in best dominated approximation problems in Banach lattices as the respective rotundity properties (strict and uniform rotundity) do in best approximation problems in Banach spaces. Moreover, monotonicity properties are applicable in ergodic theory. Recall also that they are restrictions of appropriate rotundity properties to the set of couples of comparable elements in the positive cone of a Banach ideal space \(E\). The author presents criteria for strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity, but the main result in that section concerns uniform monotonicity. It is natural that for the characterization of uniform monotonicity some regularity condition is required. The author introduces such a condition and he shows that it is equivalent to the notion of classical regularity of the weight function \(\omega \) in the special case of classical Orlicz-Lorentz sequence spaces. In the last section, the author discusses the non-squareness of the spaces \(\lambda _{\varphi }\), of their subspaces of order continuous elements \(\left( \lambda _{\varphi }\right) _{a}\) and of finite dimensional subspaces \(\lambda _{\varphi }^{n}\) of \(\lambda _{\varphi }\). The proofs concerning non-squareness differ significantly from proofs of previously discussed properties and often require new techniques.
generalized Orlicz-Lorentz sequence space
scientific article; zbMATH DE number 6154629