Dilation and Model Theory for Pairs of Commuting Contraction Operators

Presents a parallel of the Sz.-Nagy–Foias dilation and functional model theory for single contractions in the bi-variate setting.

This reference will be of interest to researchers working in multivariable operator theory. It is accessible to anyone with a first-year graduate-student-level background in analysis, in particular Hardy-space operator theory and functional analysis.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.