On disjoint function systems and complemented subspaces (Q987447)

Given a sequence \(\{x_k\}^\infty_{k=1}\) of disjoint functions in \(L_r[0,1]\) for some \(1< r\leq\infty\), define the set \(R(\{x_k\}^\infty_{k=1})\) of all scalars \(p\in(1,\infty)\) such that \[ \sup_k {\| x_k\|^{{1\over 2}}_{{p\over 1+\varepsilon p}}\| x_k\|^{{1\over 2}}_{{p\over 1-\varepsilon}}\over\| x_k\|_p}< \infty \] for some \(\varepsilon> 0\). The main result of the paper is the following. Theorem. Let \(\{x_k\}^\infty_{k=1}\) be a sequence of disjoint functions in \(L_v[0,1]\) for some \(1< v<\infty\). Let \(E[0,1]\equiv E\) be a rearrangement invariant space with \(\alpha_E, \beta_E\) the Boyd indices of \(E\). If \(0<\alpha_E= \beta_E< 1\) and \({1\over\alpha_E}\in R(\{x_k\}^\infty_{k=1})\), then the subspace \([x_k]\) is complemented in \(E\). The proof uses the fact that \(R(\{x_k\}^\infty_{k=1})\) is an open subset in \((1,\infty)\) and the Boyd interpolation theorem. This theorem is applied to the case of Lorentz spaces.