On the galb of weighted Orlicz sequence spaces. II (Q1080108)

The galb theory of topological vector spaces has been developed by \textit{Ph. Turpin} in Diss. Math. 131, 221 p. (1976; Zbl 0331.46001). The study of the galbs of weighted Orlicz sequence spaces \(\ell^{\phi}(a)\) started with a previous paper of the author [Bull. Acad. Pol. Sci., Ser. Sci. Math. 32, 193-202 (1984; Zbl 0561.46006)] showing some links among the indices and the galbs of the spaces \(\ell^{\phi}(a)\), f.i., if \(\beta^{\infty}_{\phi}\) is the upper index of \(\phi\) at \(\infty\) then the galb of \(\ell^{\phi}(a)\) is contained in \(\ell^{\beta^{\infty}_{\phi}}\) for every \((a_ n)\) with \(\sum a_ n<\infty\). In the present paper it is proved that this upper bound can be attained. More precisely, if \(\phi\) is equivalent to a \(\beta^{\infty}_{\phi}\)-concave function then there exists a sequence \((a_ n)\) with \(\sum a_ n<\infty\) such that \(\ell^{\phi}(a)\) is isomorphic to \(\ell^{\beta^{\infty}_{\phi}}\). The cases of weight sequences with \(a_ n\to \infty\) or \(\sum a_ n=\infty\) are also analyzed, comuting the exact galb for a wide class of spaces \(\ell^{\phi}(a)\). This extends a Theorem of \textit{Ph. Turpin} [Stud. Math. 46, 153-165, 167-195 (1973; Zbl 0227.46040 and Zbl 0227.46041)]. As consequence it is proved the existence of p-normed Orlicz spaces defined by non p-convex Orlicz function \(\phi\) on purely atomic probabilistic measures \((0<p\leq 1)\). This shows that the known \textit{S. Mazur, W. Orlicz} and \textit{W. Matuszewska} characterization [Stud. Math. 17, 97-119 (1958; Zbl 0085.322) and 21, 317-344 (1962; Zbl 0123.304)] fails for this class of spaces \(\ell^{\phi}(a)\). Similar phenomena appear in the case of \(\ell^{\phi}(a)\) for \(a_ n\to \infty\), too.