Complete characterizations of Kadec-Klee properties in Orlicz spaces

We study the connections between the Kadec-Klee property for local convergence in measure H-l, the Kadec-Klee property for global convergence in measure H-g and the Delta(2)-condition for Orlicz function spaces L-phi equipped with either the Luxemburg norm parallel to . parallel to(phi) or the Orlicz norm parallel to.parallel to(rho)(0). Nominally, we prove that for (L-phi, parallel to . parallel to(phi)) the conditions: phi satisfies an appropriate Delta(2)-condition and L-phi is an element of H-l, L-phi is an element of H-g are equivalent, although L-phi is an element of H-g is not equivalent to E-phi is an element of H-g. In contrast, we also prove that, in the case of a non-atomic infinite measure space, properties H-l and H-g for (L-phi, parallel to .parallel to(phi)(0)) do not coincide. More precisely, we prove that if phi vanishes only at zero, then both these properties coincide and they are equivalent to phi is an element of Delta(2). However, if phi vanishes outside zero, then (L-phi, parallel to . parallel to(phi)(0)) is an element of H-g if and only if phi is an element of Delta(2)(infinity). Since in the last case (Lphi, parallel to parallel to(phi)(0)) is not order continuous, properties H-l and H-g differ. Analogous results are also proved for the subspace E-phi of L-phi. It is also worth mentioning that the criteria for E-phi is an element of H-l as well as for E-phi is an element of H-g were not previously known. It follows from the criteria that the appropriate regularity Delta(2)-condition for phi is necessary for E-phi is an element of H-l, E-0(phi) is an element of H-l, E-phi is an element of H-g and E-0(phi) is an element of H-g although these spaces are order continuous for any phi.