Document Type


Publication Date



Let K1 and K2 be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator R0(R0f(z)=(f(z)−f(0))/z), and let A be a linear continuous operator from K1 into K2 whose adjoint commutes with R0. 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.


This article was originally published in Annales de l'Institut Fourier, volume 39, in 1989.

This article is in French.

Peer Reviewed



Association des Annales de l'Institut Fourier



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.