The best approximation by projections in Banach spaces (Q5944185)

The simple `main result' of the paper, shorn of inessential details, contains the following result. Let \(X\) be a Banach space and let \(M\) be a projection on \(X\). Let \({\mathcal L}\) be the set of projections on \(X\) with the same range as \(M\). Let \((\Omega,\mu)\) be a probability measure space and \(\{T_t: t\in \Omega\}\) and \(\{U_t: t\in\Omega\}\) be uniformly bounded families in \(B[X]\), the set of bounded linear mappings on \(X\), all commuting with \(M\), and such that \(t\to T_tTU_tf\) is strongly \(\mu\)-measurable for all \(t\in\Omega\), \(f\in X\) and \(T\in B[X]\). Then, for \(T\in B[X]\), \(\Phi_T\in B[X]\) is defined by NEWLINE\[NEWLINE\Phi_T(f)= \int_\Omega T_tTU_t(f) d\mu(t)NEWLINE\]NEWLINE and \({\mathcal L}^*= \{T\in{\mathcal L}: \Phi_T= M\}\). If \(\|\Phi\|\leq 1\), then for every \(S\in B[X]\) which commutes either with all \(T_t\) or with all \(U_t\) (and so for every scalar operator) the projection \(M\) is the best approximation to \(S\) from \({\mathcal L}^*\). Realizations of this situation are provided by \(M\) such that \(M(f)\) is a partial sum of a generalized Fourier expansion, by \(B_t\) and \(U_t\) which are multiplier operators (for some of which \({\mathcal L}^*= {\mathcal L}\)) and by certain convolution type operators \(S\).