Dilatations des commutants d'opérateurs pour des espaces de Krein de fonctions analytiques

Abstract

Let 𝒦 1 and 𝒦 2 be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator R 0 (R 0 f(z)=(f(z)-f(0))/z), and let A be a linear continuous operator from 𝒦 1 into 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-Θ(z)JΘ * (ω))/(1-zω * ) where J is a matrix subject to J=J * =J -1 and where Θ is analytic in the unit disk and J-unitary on the unit circle.