The range of a contractive projection in \(L_p(H)\).
The main purpose of the paper under review is to investigate structural aspects of ranges of (linear) contractive projections on Bochner spaces of Hilbert space valued functions. Since the results of \textit{R. G. Douglas} [Pac. J. Math. 15, 443--462 (1965; Zbl 0148.12203)] and \textit{T. Ando} [Pac. J. Math. 17, 391--405 (1966; Zbl 0192.23304)] on contractive projections on \(L_p\)-spaces, the vector-valued case has been a long standing problem. In this interesting paper, the author settles the case of Hilbert-space valued Bochner spaces. More precisely, it is shown that the range of a linear contractive projection on a Lebesgue-Bochner space of Hilbert valued functions is isometric to an \(l_p\)-direct sum of Hilbert space valued \(L_p\)-spaces. Also, the precise form of the contractive projections is given. The method of proof is very elegant and is a delicate generalization of the method that goes back to [\textit{S. J. Bernau} and \textit{H. E. Lacey}, Pac. J. Math. 53, 21-41 (1974; Zbl 0287.46038)]. One should note that the question of describing the range/form of linear contractive projections on \(L_p(X)\) remains open (even for the case of \(X=L_q\) with \(q\neq 2\)).
scientific article; zbMATH DE number 2109948