Unitary colligations of operators in Krein spaces
A colligation of operators in Hilbert spaces is a set \(\delta\) consisting of three Hilbert spaces: \({\mathcal H}_ 1\), \({\mathcal H}_ 2\) and \({\mathcal H}_ 0\), and four linear operators: \(A_{11}\in {\mathcal L}({\mathcal H}_ 0,{\mathcal H}_ 2)\), \(A_{12}\in {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\), \(A_{21}\in {\mathcal L}({\mathcal H}_ 0,{\mathcal H}_ 0)\) and \(A_{22}\in {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 0)\). A colligation \(\delta\) is said to be unitary if the operator matrix \(A=[A_{ij}]_{i,j=1,2}\) is a unitary operator from \({\mathcal H}_ 0\oplus {\mathcal H}_ 1\) to \({\mathcal H}_ 2\equiv {\mathcal H}_ 0\). When Hilbert spaces \({\mathcal H}_ 1\), \({\mathcal H}_ 2\) and \({\mathcal H}_ 0\) are replaced by Krein spaces \({\mathcal K}_ 1\), \({\mathcal K}_ 2\) and \({\mathcal K}_ 0\) and unitarity is understood in the sense of Krein space structure, we can speak of a unitary colligation of operators in Krein spaces. The characteristic function \(\Theta_{\delta}(z)\) of \(\delta\) is the \({\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\)-valued analytic function defined by \[ \Theta_{\delta}(z)=A_{12}+zA_{11}(I-zA_{21})^{-1} A_{22}. \] It is known that a \({\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\)- valued function, defined and analytic on the open unit disk \({\mathbb{D}}\) coincides with the characteristic function of a unitary colligation if and only if \(\Theta\) (z) is contractive, i.e. \(\| \Theta (z)\| \leq 1\) for all z in \({\mathbb{D}}.\) Let \({\mathcal K}_ 1\), \({\mathcal K}_ 2\) be Krein spaces and \(\Theta\) (z) a \({\mathcal L}({\mathcal K}_ 1,{\mathcal K}_ 2)\)-valued bounded function, defined and analytic on \({\mathbb{D}}\). The author constructs a unitary colligation of operators in Krein spaces whose characteristic function coincides with \(\Theta\) (z). Moreover he gives a new proof for a McEnnis' characterization for \(\Theta\) (z) to be written in the form \[ \Theta (z)=\{-TH_ T+zD_{T^*}(I-zT^*)^{-1} D_ T\}|_{{\mathcal D}_ T}, \] where \(D_ T=| I-T^*T|^{1/2}\), \(J_ T=sgn(I-T^*T)\) and \({\mathcal D}_ T\) is the closure of the range of \(D_ T\).
unitary colligation of operators in Krein spaces
scientific article; zbMATH DE number 4165801
scientific article