Contractions with Polynomial Characteristic Functions II. Analytic Approach

TL;DR

This paper characterizes completely nonunitary contractions with polynomial characteristic functions as operators related to nilpotent operators, providing a structural decomposition involving coisometries and isometries.

Contribution

It proves a structural theorem linking polynomial characteristic functions of contractions to nilpotent operators and explicit isometric and coisometric mappings.

Findings

Contractions with polynomial characteristic functions are structurally related to nilpotent operators.

The characteristic function admits a specific factorization involving isometries and coisometries.

Provides a classification framework for such contractions based on polynomial degree.

Abstract

The simplest and most natural examples of completely nonunitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions are the nilpotent operators. The main purpose of this paper is to prove the following theorem: Let $T$ be a completely nonunitary contraction on a Hilbert space $\mathcal{H}$. If the characteristic function $\Theta_T$ of $T$ is a polynomial of degree $m$, then there exist a Hilbert space $\mathcal{M}$, a nilpotent operator $N$ of order $m$, a coisometry $V_1 \in \mathcal{L}(\overline{ran} (I - N N^*) \oplus \mathcal{M}, \overline{ran} (I - T T^*))$, and an isometry $V_2 \in \mathcal{L}(\overline{ran} (I - T^* T), \overline{ran} (I - N^* N) \oplus \mathcal{M})$, such that \[ \Theta_T = V_1 \begin{bmatrix} \Theta_N & 0 0 & I_{\mathcal{M}} \end{bmatrix} V_2. \]