Isomorphic copies in the lattice E and its symmetrization with applications to Orlicz–Lorentz spaces

Abstract: The paper is devoted to the isomorphic structure of symmetrizations of quasi-Banach ideal function or sequence lattices. The symmetrization of a quasi-Banach ideal lattice E of measurable functions on , , or , consists of all functions with decreasing rearrangement belonging to E. For an order continuous E we show that every subsymmetric basic sequence in which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements in E, and conversely. Among several consequences we show that, provided E is order continuous with Fatou property, contains an order isomorphic copy of if and only if either E contains a normalized -basic sequence which converges to zero in measure, or contains the function . We apply these results to the family of two-weighted Orlicz–Lorentz spaces defined on or , . This family contains usual Orlicz–Lorentz spaces when and Orlicz–Marcinkiewicz spaces when . We show that for a large class of weights , it is equivalent for the space , and for the non-weighted Orlicz space to contain a given sequential Orlicz space isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of in these spaces over or exclusively in terms of the indices of φ. If we show that the set of exponents p for which lattice embeds in the order continuous part of is the union of three intervals determined respectively by the indices of φ and by the condition that the function belongs to the space. [Copyright &y& Elsevier]