Unbounded symmetrizable idempotents
Let \({\mathcal H}\) be a Hilbert space, \({\mathcal S}\) a closed subspace of \({\mathcal H}\) and \(A\) a positive (semidefinite bounded) operator acting on \({\mathcal H}\). A closed densely defined projection \(Q\) is called \(A\)-symmetric if \(AQ\) is symmetric, and a pair \( ({\mathcal S}, A)\) is called quasi-compatible if there exists an \(A\)-symmetric projection \(Q\) with range \({\mathcal S} \); under the additional assumption that \(Q\) is bounded, that pair is called compatible.NEWLINENEWLINEFrom the introduction: ``Section 1 contains some preliminaries and a quite complete description of all pairs of closed subspaces with some properties which appear in the different compatibility notions. Section 2 is devoted to describing unbounded projections which are \(A\)-symmetric for a given positive bounded operator \(A\). We show that injectivity or invertibility of \(P_{1}+P_{2}\) and \(P_{1} - P_{2}\) or properties of \(\|P_{1}+P_{2}-I\|\) or \(\|P_{1}P_{2}\| \) (for a convenient choice, in each case, of orthogonal projections \(P_{1}\) and \(P_{2}\)) give equivalent conditions for the compatibility and quasi-compatibility of a given pair \( ({\mathcal S}, A) \). In addition, we describe the set \({\mathcal P}(A, {\mathcal S})\) of all \(A\)-symmetric projections with range \({\mathcal S}\). We also prove the existence of a distinguished \(P_{A,{\mathcal S}}\in {\mathcal P}(A, {\mathcal S})\) with several nice properties. Section 3 shows how to calculate, more or less explicitly, the projection \(P_{A,{\mathcal S}}\). Finally, Section 4 contains applications of quasi-compatibility to abstract interpolation, splines and least squares problems.''
scientific article; zbMATH DE number 6044814
scientific article