We consider a Krein space isometry whose defect subspaces are Hilbert spaces and we show that its minimal unitary Hilbert space extensions are related to one-step isometric Hilbert space extensions and Schur parameters. These unitary extensions give rise to moments and scattering matrices defined on a scale subspace. By means of these notions we solve the labeling problem for the contractive intertwining liftings in the commutant lifting theorem for Krein space contractions.
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Aad Dijksma, S.A.M. Marcantognini, H.S.V. de Snoo, A Schur Type Analysis of the Minimal Unitary Hubert Space Extensions of a Krein Space Isometry whose Defect Subspaces are Hubert Spaces. Z. Anal. Anwend. 13 (1994), no. 2, pp. 233–260