Contractions with polynomial characteristic functions I. Geometric approach

Abstract

In this paper we study the completely nonunitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form \begin{equation*} T = \begin {bmatrix}S & * & *\\0 & N & *\\0& 0& C \end{bmatrix}, \end{equation*} where $S$ and $C^*$ are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that the dimension of ker$S^*$ and the dimension of $\mbox {ker} C$ 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.

Locations

• arXiv (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View - PDF