Contractions with polynomial characteristic functions. I: Geometric approach (Q2841347)

This is a nice contribution to the well-known theory of Hilbert space contractions due to B. Sz.-Nagy and the first author. In this paper, it is proved that the characteristic function of a completely nonunitary contraction \(T\) is a polynomial if and only if \(T\) has a matrix representation of the form \({\begin{bmatrix} S&*&*\\0&N&*\\0&0&C\end{bmatrix}}\), where \(S,C^*\) are unilateral shifts and \(N\) is nilpotent. First, the authors study an illustrative example for which \(S\) and \(C^*\) are unilateral shifts of multiplicity 1. For the general case, they prove that the multiplicities of \(S\) and \(C^*\) are unitary invariants of \(T\) and \(N\) is, up to a quasi-similarity, uniquely determined by \(T\). Moreover, contractions for which their characteristic functions are monomials are completely described in this paper.