A Schur Analysis of the Minimal Weak Unitary Dilations of a Contraction Operator and the Relaxed Commutant Lifting Theorem

Abstract

A Schur-type analysis of the minimal weak unitary dilations of a given contraction operator is obtained from the Arov-Grossman functional model. The result is combined with the coupling method to give a description of the interpolants in the Relaxed Commutant Lifting Theorem. (2000). Primary 47A20; Secondary 47A57, 47A56. Mathematics Subject Classification

A Schur type analysis of the minimal weak unitary Hilbert space dilations of a Kreĭn space bicontraction and the Relaxed Commutant Lifting Theorem in a Kreĭn space setting

“…Kaashoek in [13] and [14], by W.S. Li and D. Timotin in [15], and by the authors of the present note in [17].…”

“…Arov and L.Z. Grossman [2,3] in the Hilbert space framework (see also [17]). A more general result for continuous isometric operators with regular domain and range is given in [7].…”

“…We claim that W ∈ L( A) and U ϑ ∈ L(F ϑ ) are undistinguishable elements of WUD(X). The proof is obtained by mimicking the arguments in [17].…”

“…We follow the approach adopted in [17]: first we present a version of the Arov-Grossman model which provides a parameterization of the minimal unitary Hilbert space extensions of a given Kreȋn space isometry whose defect subspaces are Hilbert spaces, then we use it to parameterize the minimal weak unitary Hilbert space dilations of a given continuous bicontraction X defined on a regular subspace B of a Kreȋn space H such that H B is a Hilbert subspace of H, finally we tackle the relaxed commutant lifting problem with a given data set {C, T , V T , R, Q} of five Kreȋn space operators, where T is a continuous bicontraction, and give a parametric descriptions of the interpolants in this setting. In addressing the lifting problem we apply the coupling method to get a continuous bicontraction X from the data set {C, T , V T , R, Q} and we show that the interpolants can be obtained from a subclass of the minimal weak unitary Hilbert space dilations of X.…”