Journal of Operator Theory

Volume 78, Issue 2, Fall 2017 pp. 281-291.

Contractions with polynomial characteristic functions. II. Analytic approach

Authors: Ciprian Foias (1), Carl Pearcy (2), and Jaydeb Sarkar (3)
Author institution:(1) Department of Mathematics, Texas A $\&$ M University, College Station, Texas 77843, U.S.A.
(2) Department of Mathematics, Texas A $\&$ M University, College Station, Texas 77843, U.S.A.
(3) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

Summary: 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 $V_1 \in \mathcal{L}(\overline{\mbox{ran}} (I - N N^*) \oplus \mathcal{M}, \overline{\mbox{ran}} (I - T T^*))$, and an isometry $V_2 \in \mathcal{L}(\overline{\mbox{ran}} (I - T^* T), \overline{\mbox{ran}} (I - N^* N) \oplus \mathcal{M})$, such that \[ \Theta_T = V_1 \begin{bmatrix} \Theta_N & 0 \\ 0 & I_{\mathcal{M}} \end{bmatrix} V_2. \]

DOI: http://dx.doi.org/10.7900/jot.2016aug11.2146
Keywords: characteristic function, model, nilpotent operators, operator valued polynomials

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