Contractions with Polynomial Characteristic Functions II. Analytic Approach

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 $$\Theta_T = V_1 \begin{bmatrix} \Theta_N & 0 0 & I_{\mathcal{M}} \end{bmatrix} V_2.$$