Contractions with Polynomial characteristic functions I. Geometric approach

TL;DR

This paper classifies a special class of non-unitary contractions with polynomial characteristic functions on Hilbert spaces, using a geometric approach to understand their structure and invariants.

Contribution

It provides a new geometric framework for classifying contractions with polynomial characteristic functions, identifying key invariants and a complete subclass characterization.

Findings

Dimension of ker S* and ker C are unitary invariants.

Nilpotent part N is uniquely determined up to quasi-similarity.

Complete classification for contractions with monomial characteristic functions.

Abstract

In this note we study the completely non unitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form T = S & * & * 0 & N & * 0& 0& C, where $S$ and C^* are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that dimension of ker S^* and dimension of ker C are unitary invariants of $T$ and that N, up to a quasi-similarity is uniquely determined by T. Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.