Dilatations des commutants d'opérateurs pour des espaces de Krein de fonctions analytiques. (Dilations of commutants of operators for Krein spaces of analytic functions)
Let \({\mathcal K}_ 1\) and \({\mathcal K}_ 2\) be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator \(R_ 0\) \((R_ 0f(z)=(f(z)-f(0))/z)\), and let A be a linear continuous operator from \({\mathcal K}_ 1\) into \({\mathcal K}_ 2\) whose adjoint commutes with \(R_ 0\). We study dilations of A which preserve this commuting property and such that the Hermitian forms defined by \(I-AA^*\) and \(I- BB^*\) have the same number of negative squares. We thus obtain a version of the commutant lifting theorem in the framework of Krein spaces of analytic functions. To prove this result we suppose that the graph of the operator \(A^*\), in the metric defined by \(I-AA^*\), is a reproducing kernel Pontryagin space of analytic functions whose reproducing kernel is of the form \((J-\Theta (z)J\Theta^*(\omega))/(1- z\omega^*)\) where J is a matrix subject to \(J=J^*=J^{-1}\) and where \(\Theta\) is analytic in the unit disk and J-unitary on the unit circle.
commutant lifting theorem in the framework of Krein spaces of analytic functions
reproducing kernel Pontryagin space of analytic functions
scientific article; zbMATH DE number 4103732
scientific article