A constructive approach to minimal projections in Banach spaces
Let X be a Banach space, Y a finite dimensional subspace of X and \({\mathcal P}(X,Y)\) be the family of all projections of X onto Y. The projection \(P\in {\mathcal P}(X,Y)\) will be called a minimal projection if \(\| P\| \leq \| Q\|\) for all \(Q\in {\mathcal P}(X,Y)\). In order to compute the norm of the minimal projections, the author considers the sequences of finite dimensional ''approximating subspaces'' of X, \(\{X_ m\}\) and \(\{Y_ m\}\) and the corresponding minimal projections \(P_ m: X_ m\to Y_ m\) such that \(\lim_{m\to \infty}\| P_ m\| =\| P\|.\) One shows that such sequences can be constructed in many important Banach spaces including C[a,b], \(L^ p[a,b]\) and \(\ell^ p\) for \(1\leq p<\infty\) and \(c_ 0\). In such spaces, sequences of projections onto Y can also be constructed, having minimal projections as cluster points (in the strong operator topology). As an application, \(\inf \{\| Q\|: Q\in {\mathcal P}(X,Y)\}\) is computed in some particular cases.
scientific article; zbMATH DE number 3960215
scientific article