Linearity of the nearest point cross section operators on holomorphic function spaces (Q1899789)

Let \(X\) be a closed subspace of a Banach space \(Y\). A map \(M\) (which in general is nonlinear) from \(Y/X\) into \(Y\) is said to be the nearest point cross section, if \(M(w) \in w\) and \(\| w\|_{Y/X}=\| w \|_{Y}\) for every \(w \in Y/X\). While in general the nearest point cross section need not exist, in case of the classical Hardy space \(H^1\), the nearest point cross section provides an efficacious tool for characterizing weakly compact subsets of \(L^1/H^1_0\) and weakly compact subsets of the dual of the disk algebra [see \textit{A. Pełczyński}, Regional Conference Series in Mathematics. No. 30. Providence, R. I.: Am. Math. Soc. (1977; Zbl 0384.46015)]. In the paper under review the authors study the case when the space \(Y\) is the Bergman space \(L^ p_a(\Omega)\) (\(1 < p <\infty\)), where \(\Omega\) is a bounded domain in \({\mathbb{C}}^n\) and the vectors are holomorphic functions (in \(\Omega\)) which belong to \(L^p\) with respect to Lebesgue measure on the domain. The main results (cf. Theorem 1 and Theorem 2) of the paper are too technical to cite here in detail. As a corollary of the main theorems the authors obtain a necessary and sufficient condition for a domain in \({\mathbb{C}}^ n\), which is holomorphically equivalent to the unit ball of \({\mathbb{C}}^ n\), to contain subsets invariant for certain minimal interpolation. The paper concludes with an example which illustrates how to use shear mappings of \({\mathbb{C}}^ n\) to construct such a domain.