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.
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.
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