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arXiv · 1604.05485

Contractions with Polynomial Characteristic Functions II. Analytic Approach

Ciprian Foias·Carl Pearcy·Jaydeb Sarkar
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Abstract

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 $Θ_T$ 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 \[ Θ_T = V_1 \begin{bmatrix} Θ_N & 0 0 & I_{\mathcal{M}} \end{bmatrix} V_2. \]

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Ciprian Foias, Carl Pearcy, Jaydeb Sarkar. 2017-04-19. Contractions with Polynomial Characteristic Functions II. Analytic Approach. https://arxiv.org/abs/1604.05485

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