Weighted projections into closed subspaces (Q2840503)

Let \(L(\mathcal{H})\) be the algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\) and \(L(\mathcal{H})^{+}\) be the cone of semidefinite positive operators of \(L(\mathcal{H})\) . In the paper under review, the notion of the compatibility of a pair \((A,\mathcal{S})\) of an operator \(A\in L(\mathcal{H})^{+}\) and a closed subspace \(\mathcal{S} \subseteq \mathcal{H}\) plays a fundamental role; this notion was introduced by the first author et al. [Acta Sci. Math. 67, No. 1--2, 337--356 (2001; Zbl 0980.47021); Linear Algebra Appl. 341, No. 1--3, 259--272 (2002; Zbl 1015.47014); J. Approx. Theory 117, No. 2, 189--206 (2002; Zbl 1055.47016)] and it means that \(\mathcal{H}= \mathcal{S} + A(\mathcal{S})^{\perp}\).NEWLINENEWLINEThe main goal of the paper is to establish the relationship between compatibility and general weighted least squares problems that were first considered for finite-dimensional Hilbert spaces by \textit{S. K. Mitra} and \textit{C. R. Rao} [Linear Algebra Appl. 9, 155--167 (1974; Zbl 0296.15002)]. In particular, there was defined a so-called \(A\)-weighted least squares process, or \(A\)-weighted projection into \( \mathcal{S} \) with respect to a seminorm \(\|\bullet\|_{A} = \|A^{1/2}\bullet \|\), as such an operator \(T\in L(\mathcal{H})\) that \(T(\mathcal{H}) \subseteq \mathcal{S}\) and \(\|y-Ty\|_{A}\leq \|y-s\|_{A}\) for all \(y \in \mathcal{H}\) and all \(s \in \mathcal{S}\).