Characteristic Functions, Liftings and Modules

TL;DR

This paper develops a comprehensive theory of characteristic functions for row contractions, linking them to modules, liftings, and completely positive maps, with applications to invariant subspaces and fixed point sets.

Contribution

It introduces a new notion of characteristic function for ergodic and coisometric row contractions, extending the analysis to contractive liftings and module structures.

Findings

Characteristic functions serve as complete invariants for certain row contractions.

The theory connects characteristic functions with fixed point sets of completely positive maps.

Applications include analysis of constrained row contractions and module structures.

Abstract

We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications to fixed point sets of completely positive maps related to each other by a subisometric lifting. In the last two chapter we consider constrained row contractions and some module structures in this context.