Symmetric relations and module spaces
\(S\) is said to be a closed linear relation in a Hilbert space \(\mathfrak R\) if \(S\) is a closed linear subspace of the orthogonal direct sum \({\mathfrak R}\oplus {\mathfrak R}\). If \(S\) is a graph of a linear operator then it is identified with this operator. Write: \(\Lambda(S)= \{\ell\in \mathbb{C}\mid (S-\ell)^{-1}\) is a bounded linear operator\}, where \[ (S- \ell)^{- 1}= \bigl\{\{g- \ell f,f\}\in {\mathfrak R}\oplus {\mathfrak R}\mid \{f,g\}\in S\bigr\}. \] Let \(\mathfrak K\) be a non-trivial closed subspace of \(\mathfrak R\). If \[ \Omega({\mathfrak K})= \bigl\{\ell\in \Lambda(S)\mid {\mathfrak R}=\text{ran}(S- \ell)^{-1}+ {\mathfrak K},\text{ direct sum}\bigr\}\neq\emptyset, \] then \(\mathfrak K\) is said to be a module space. Several properties of module spaces are given without the requirement that defect spaces are finite-dimensional. Particular attention is paid to module spaces for symmetric relations.
scientific article; zbMATH DE number 604800