On series with respect to the Rademacher system in rearrangement invariant spaces ``close'' to \(L_\infty\)
The author presents some results on the behavior of series with respect to the Rademacher system (\(r_k\)) in a r.i.i. space, i.e. a rearrangement invariant Banach space of measurable functions on \([0,1]\) that is intermediate between \(L_\infty\) and \(G\) (the closure of \(L_\infty\) in the Orlicz space \(L_M\), where \(M(t)= \exp(t^2)- 1\), \(t\in [0,1]\)) [see the classical book of \textit{S. G. Krein}, \textit{Yu. I. Petunin} and \textit{E. M. Semenov}, ``Interpolation of linear operators'' (in Russian), Moscow (1978; Zbl 0499.46044)]. The following results concerning interpolation spaces are announced: I. If \(X\) is a r.i.i. space, then \[ \Biggl\|\sum^\infty_{k=1} a_kr_k\Biggr\|_X= \|(a_k)\|_F,\quad\text{where } F= (l_1l_2)^K_E, \] for a parameter \(E\) of \(X\) given by the real \(K\)-method of \textit{N. J. Kalton} [Stud. Math. 106, No. 3, 233-277 (1993; Zbl 0810.46020)]. II. Let \(X_0\), \(X_1\) be two r.i.i. spaces. If \[ \Biggl\|\sum^\infty_{k=1} a_kr_k\Biggr\|_{X_0}= \Biggl\|\sum^\infty_{k=1} a_kr_k\Biggr\|_{X_1}, \] then \(X_0= X_1\) and the norms in \(X_0\) and \(X_1\) are equivalent. Similar results for sequence spaces satisfying more restrictive conditions have been obtained by \textit{V. A. Rodin} and \textit{E. M. Semenov} [Analysis Math. 1, 207-222 (1975; Zbl 0315.46031)].
scientific article; zbMATH DE number 1292500