The strong property (B) for \(L_p\) spaces
Answering a question by \textit{R. W. Cross} et al. [Math. Nachr. 173, 91--114 (1995; Zbl 0834.47001)] (see also [\textit{A.~Martínez-Abejón} et al., Mat. Stud. 24, No.~2, 173--191 (2005; Zbl 1094.46502)]), the author proves that for every infinite dimensional subspace \(X\) of \(L_p(\nu)\) for a finite atomless \(\nu\) and \(1 \le p < \infty\) there exists a decomposition \(L_p(\nu) = Y \oplus Z\) into subspaces \(Y\) and \(Z\) both isomorphic to \(L_p(\nu)\) such that both subspaces \(X \cap Y\) and \(X \cap Z\) are infinite dimensional. The proof is pretty involved. As a consequence, he extends the following result by the aforementioned authors [\textit{R. W. Cross} et al., Math. Nachr. 173, 91--114 (1995; Zbl 0834.47001)] from \(1 < p < \infty\) to \(1 \le p < \infty\). Given a Banach space \(Y\) and a linear bounded operator \(T \colon L_p \to Y\) with a dense non-closed range \(T(L_p)\) in \(Y\), there exists a pair of linear bounded operators \(T_i \colon L_p \to Y\), \(i = 1,2\), with the following properties. (i) \(T_i(L_p)\) is dense and non-closed in \(Y\), \(i = 1,2\). (ii) \(T(L_p) = T_1(L_p) + T_2(L_p)\). (iii) \(T_1(L_p) \cap T_2(L_p) = \{0\}\).
scientific article; zbMATH DE number 7435776
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