Dilatations Des Commutants D'Opérateurs Pour Des Espaces De Krein De Fonctions Analytiques

Authors

Daniel Alpay, Chapman UniversityFollow

Document Type

Article

Publication Date

1989

Abstract

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.

Comments

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

This article is in French.

Recommended Citation

D. Alpay, Dilatations des commutants d'opérateurs pour des espaces de Krein de fonctions analytiques. Annales de l'Institut Fourier, vol. 39 (1989), pp. 1073-1094.

Peer Reviewed

1

Copyright

Association des Annales de l'Institut Fourier