Characteristic function of a power partial isometry

TL;DR

This paper classifies purely contractive analytic functions whose associated contractions are c.n.u. power partial isometries, and characterizes operator-valued symbols for Toeplitz partial isometries, extending Sz.-Nagy-Foia ext{s} theory.

Contribution

It provides a complete classification of certain contractions and characterizes operator-valued symbols for Toeplitz partial isometries, expanding the understanding of power partial isometries.

Findings

Classified purely contractive analytic functions with c.n.u. power partial isometries.

Identified contractive polynomials leading to explicit diagonal forms of contractions.

Characterized operator-valued symbols for Toeplitz operators that are partial isometries.

Abstract

The celebrated Sz.-Nagy-Foia\c{s} model theory says that there is a bijection between the class of purely contractive analytic functions and the class of completely non-unitary (c.n.u.) contractions modulo unitary equivalence. In this paper we provide a complete classification of the purely contractive analytic functions such that the associated contraction is a c.n.u. power partial isometry. As an application of our findings, we determine a class of contractive polynomials such that the associated c.n.u. power partial isometry is of the explicit diagonal form $S \oplus N \oplus C$, where $S$ and $C^*$ are unilateral shifts and $N$ is nilpotent. Finally, we obtain a characterization of operator-valued symbols for which the corresponding Toeplitz operator on vector-valued Hardy space is a partial isometry.