Contractions with polynomial characteristic functions I. Geometric approach

Foias, Ciprian; Sarkar, Jaydeb

doi:10.1090/S0002-9947-2012-05450-X

Contractions with polynomial characteristic functions I. Geometric approach
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by Ciprian Foias and Jaydeb Sarkar PDF

Trans. Amer. Math. Soc. 364 (2012), 4127-4153 Request permission

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.

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