Compressions of Linear Relations in Krein Spaces

Given an orthogonal decomposition \({H} = H[]{H}\) of a Krein space, we study compressions \({CH}(R)\) of linear relations \(R\) in \({H}\) to \(H\). We formulate a sufficient condition that ensures that compression commutes with taking adjoints, that is, the adjoint identity \({CH}(R^{{*}}) = {CH}(R)*\) 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 \(H≠\{0\}\), we characterize finite dimensionality of the exit space \({H}\) by universal compression properties. Finally, we give a criterion for the adjoint identity to hold in terms of bounded linear functionals on \(R\).