Universal Musielak-Orlicz sequence spaces (Q2725626)

Let \(\mathcal O\) be the set of all Orlicz functions (equipped with the topology of uniform convergence on compact sets), \({\mathcal A}\) be the set of all sequences \(F=(f_n)\) in \(\mathcal O,\) and \(\ell_{\mathcal A}\) be the class of all Musielak-Orlicz sequence spaces \(\ell_F\) \((F\in {\mathcal A}).\) Essentially, the paper contains three results on \(F,G\in {\mathcal A}\) and the corresponding sequence spaces \(\ell_F\) and \(\ell_G\) (cf. Theorem 1 and 2): If \(F\in {\mathcal A}\) is dense in \(\mathcal O,\) then \(F\) is universal for \({\mathcal A}\) and \(\ell_F\) is perfectly complementably universal for \(\ell_{\mathcal A}.\) If \(F\) and \(G\) are universal for \({\mathcal A},\) then they are permutatively equivalent and \(\ell_F\) and \(\ell_G\) are isomorphic. If both \(F\) and \(G\) are dense in \(\mathcal O,\) then they are permutatively equivalent and \(\ell_F\) and \(\ell_G\) are nearly isometric.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].