A Hilbert space characterization among function spaces
A Banach space \(E\) of measurable functions on the interval \([0,1]\) is said to be rearrangement invariant provided that, (i) if \(|x(t)|\leq |y(t)|\) and \(y\) is in \(E\), then \(x\) is in \(E\) and \(|x|_E\leq|y|_E\), and, (ii) if \(x\) and \(y\) are equimeasurable functions and \(y\) is in \(E\), then \(x\) is in \(E\) and \(|x|_E=|y|_E\). Some examples of rearrangement invariant spaces (RI-spaces, for short) are the \(L_p\) and Orlicz spaces. Indeed, the inclusion \(L_\infty\subset E\subset L_1\) holds for any non-trivial RI-space \(E\), and \(|x|_1\leq|x|_E\leq|x|_\infty\) provided the norm on \(E\) is properly normalized. The projection \(P\) onto the constant functions, defined by \((Px)(t)=\int^1_0x(s)ds\) is a norm one operator from \(L_1\) into \(L_\infty\) and, hence, is also a bounded operator on every RI-space \(E\). One has \(|P|_E=1\) and \(1\leq|I-P|_E\leq 2\), where \(I\) is the identity operator on \(E\). The main result (Theorem 1) of the paper under review is that, if \(E\) is an RI-space for which \(|I-P|_E=1\), then \(E=L_2[0,1]\) and \(|x|_E=|x|_2\) for every \(x\) in \(E\). Notable among the auxiliary results used to prove the main theorem and Theorem 2, which says that, if \(X\) is a normed space of dimension at least 3 and if every triangle in \(X\) can be strictly separated from any closed bounded convex set disjoint from it by a hyperplane \(V\) for which there exists a norm one operator \(L:X\to X\) whose restriction to \(V\) is the identity, then \(X\) is a Hilbert space, and Lemma 5, which says that, if \(\{x_n\}\) is a bounded sequence in \(L_\infty\) with \(|x_n|_\infty\leq 1\) for all \(n\) and if \(E\) is a separable RI-space, then there is a subsequence \(n_k\) and some \(x_0\) in \(L_\infty\) with \(|x_0|_\infty\leq 1\) such that \(\{x_{n_k}\}\) converges to \(x_0\) weakly in \(E\). Along the way, the authors evoke a characterization of Hilbert spaces due to Garkavi and Klee, a convexity theorem of Lyapunov, and a reflexivity theorem of Lozanovskii.
scientific article; zbMATH DE number 808260