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Full Description
Exactly a decade after the publication of the Sz.-Nagy Dilation Theorem, Tsuyoshi Andô proved that, just like for a single contractive operator, every commuting pair of Hilbert-space contractions can be lifted to a commuting isometric pair. Although the inspiration for Andô's proof comes from the elegant construction of Schäffer for the single-variable case, his proof did not shed much light on the explicit nature of the dilation operators and the dilation space as did the original Schäffer and Douglas constructions for a single contraction. Consequently, there has been little follow-up in the direction of a more systematic extension of the Sz.-Nagy-Foias dilation and model theory to the bi-variate setting. Sixty years since the appearance of Andô's first step comes this thorough systematic treatment of a dilation and model theory for pairs of commuting contractions.
Contents
Preface; 1. Introduction; 2. Models for unitary dilations and isometric lifts of a contraction operator; 3. The Berger-Coburn-Lebow and Bercovici-Douglas-Foias models for pairs of commuting isometries; 4. Andô's dilation and commutant lifting theorems; 5. Douglas-type model for Andô isometric lifts; 6. Schäffer-type model for Andô isometric lifts; 7. Strongly minimal Andô isometric lifts and fundamental operators; 8. Pseudo-commuting contractive lifts; 9. Functional model and invariants for commuting contractive pairs; Appendix. More general domains and open problems; References; Index.
