Characteristic matrix of a closed linear relation (Q522977)

Let \(\mathcal H\) and \(\mathcal K\) denote complex Hilbert spaces. A \textit{linear relation} \(T\) from \(\mathcal H\) to \(\mathcal K\) is a subspace of \({\mathcal H} \times {\mathcal K}\) and is called \textit{closed} if this subspace is closed in \({\mathcal H} \times {\mathcal K}\). The orthogonal projection of \({\mathcal H} \times {\mathcal K}\) onto a closed linear relation \(T\) gives rise to a \(2 \times 2\) block operator matrix. Such a matrix is called the \textit{characteristic matrix} of \(T\). The author obtains a formula for the characteristic matrix of a closed linear relation in terms of its Moore-Penrose inverse and of its orthogonal operator part.