Compressions of Linear Relations in Krein Spaces

Abstract

Given an orthogonal decomposition \(\widetilde{\mathfrak{H}} = \mathfrak{H}[\dotplus]\widehat{\mathfrak{H}}\) of a Krein space, we study compressions \({\mathrm{C}_{\mathfrak H}}(R)\) of linear relations \(R\) in \(\widetilde{\mathfrak{H}}\) to \(\mathfrak{H}\). We formulate a sufficient condition that ensures that compression commutes with taking adjoints, that is, the adjoint identity \({\mathrm{C}_{\mathfrak H}}(R^{\widetilde{*}}) = {\mathrm{C}_{\mathfrak H}}(R)^{*}\) holds. We derive preservation and reflection results for the compression of maximal dissipative, maximal symmetric, and self-adjoint linear relations. For nonnegative linear relations, the adjoint identity also implies that compression commutes with taking the Friedrichs extension. For \(\mathfrak{H}\ne\{0\}\), we characterize finite dimensionality of the exit space \(\widehat{\mathfrak{H}}\) by universal compression properties. Finally, we give a criterion for the adjoint identity to hold in terms of bounded linear functionals on \(R\).