Weighted sequence subspaces of Orlicz function spaces isomorphic to \(\ell ^ p\)
The theory of subspaces of Orlicz sequence spaces have been developped in \textit{K. Lindberg} [Stud. Math. 45, 119-146 (1973; Zbl 0217.166)], \textit{J. Lindenstrauss}, \textit{L. Tzafriri} [Isr. J. Math. 11, 355-379 (1972; Zbl 0237.46034); 14, 368-389 (1973; Zbl 0262.46031)] and \textit{N. Kalton} [Math. Proc. Cambridge Phil. Soc. 81, 253-277 (1977; Zbl 0345.46013)], characterizing the \(\ell ^ p\)-spaces that can be embedded into an Orlicz sequence space \(\ell ^{\phi}\) in terms of the associated indices. However, the problem of determining the \(\ell ^ p\)- complemented subspaces in an Orlicz (function or sequence) space remains open. In this paper we study the Orlicz sequence spaces \(\ell ^{\psi}\) which can be isomorphically embedded into an Orlicz function space \(L^{\phi}\) via an isomorphic representation between \(\ell ^{\psi}\) and weighted Orlicz sequence spaces \(\ell ^{\phi}(w)\) with \(\sum w_ n<\infty\). (For convex functions \(\phi\), these spaces \(\ell ^{\phi}(w)\) are complemented subspaces of \(L^{\phi}).\) In particular we study the case of \(\ell ^{\psi}\equiv \ell ^ p\), \(0<p<\infty\), giving a characterization and suitable criteria to the representation \(\ell ^ p\approx \ell ^{\phi}(w)\subseteq L^{\phi}\) (the so-called \(W^ p\)-property). It is analyzed the set \(P^{\infty}_{\phi}\) of the reals \(p>0\) such that \(\phi\) has the \(W^ p\)-property, showing that it can range from being the whole associated interval \([\alpha ^{\infty}_{\phi},\beta ^{\infty}_{\phi}]\) to being the empty set. The behavior of minimal Orlicz spaces \(\ell ^{\phi}(w)\) with \(\sum w_ n<\infty\) is also studied: while in the convex case all the spaces \(\ell ^{\phi}(w)\) are isomorphic, in the non-convex case there exist uncountably many spaces \(\ell ^{\phi}(w)\) non-mutually isomorphic. As a consequence a question in \textit{F. L. Hernández} [Bull. Pol. Acad. Sci., Math. 32, 193-202 (1984; Zbl 0561.46006)] is anwered: the upper bound \(\ell ^{\beta ^{\infty}_{\phi}}\) of the galbs of the spaces \(\ell ^{\phi}(w)\) cannot be reached.
scientific article; zbMATH DE number 4009070