Dilation and Model Theory for Pairs of Commuting Contraction Operators

Exactly a decade after the publication of the Sz.-Nagy dilation theorem, Tsuyoshi Ando 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 Ando'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 bivariate setting. Sixty years since the appearance of Ando's first step comes this thorough systematic treatment of a dilation and model theory for pairs of commuting contractions.

• Provides a thorough review from scratch of the three proofs of the Sz.-Nagy dilation theorem by Schäffer, Douglas, and Sz.-Nagy–Foias, and of the model theory for isometric pairs by Berger–Coburn–Lebow and Bercovici–Douglas–Foias, including new results
• Includes two new elaborate proofs of Ando's Dilation Theorem making use of bi-variate versions of the models of Douglas and Schäffer, respectively
• Introduces an appropriate parallel of the Sz.-Nagy–Foias functional model theory for pairs of commuting contractions, including a complete set of unitary invariants for pairs of commuting contractions