On a theorem of Kadets and Pełczyński (Q2282850)

Let \(X\) denote a rearrangement invariant Banach space of measurable functions on the unit interval \([0,1]\). This paper provides conditions on \(X\) so that given a normalized unconditional basic sequence \((u_n)\) in \(X\) which is semi-normalized in \(L_1\) (i.e., \(0<\inf_n \|u_n\|_{L_1}\) and \(\sup_n\|u_n\|_{L_1}<\infty\)), the norms of \(X\) and \(L_1\) are equivalent on \([u_n]\), the closed linear span of \((u_n)\) in \(X\). This fact was proved for \(X=L_p\) (\(2<p<\infty\)) by \textit{M. I. Kadets} and \textit{A. Pełczyński} [Stud. Math. 21, 161--176 (1962; Zbl 0102.32202)]. For example, denoting by \(G\) the closure of \(L_\infty\) in the Orlicz space \(L_N\) with \(N(t)= e^{t^2}-1\), it is proved in Theorem 1 that, if \(X\supset G\), then the following conditions are equivalent: \begin{itemize} \item[(i)] For each normalized basic sequence \((u_n)\) in \(X\) equivalent to the unit vector basis of \(\ell_2\) and semi-normalized in \(L_1\), \(\|\cdot\|_X\) and \(\|\cdot\|_{L_1}\) are equivalent on \([u_n]\). \item[(ii)] \(X\) contains no disjoint sequences equivalent to the unit vector basis of \(\ell_2\). \end{itemize}