Contractions with polynomial characteristic functions I. Geometric approach

Authors:
Ciprian Foias and Jaydeb Sarkar

Journal:
Trans. Amer. Math. Soc. **364** (2012), 4127-4153

MSC (2010):
Primary 47A45, 47A20, 47A48, 47A56

DOI:
https://doi.org/10.1090/S0002-9947-2012-05450-X

Published electronically:
March 13, 2012

MathSciNet review:
2912448

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Ciprian Foias**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

**Jaydeb Sarkar**

Affiliation:
Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249

Address at time of publication:
Indian Statistical Institute, Stat-Math Unit, 8-th Mile, Mysore Road, RVCE Post, Bangalore 560059, Karnataka, India

MR Author ID:
773222

Email:
jay@isibang.ac.in, jaydeb@gmail.com

Keywords:
Characteristic function,
model,
weighted shifts,
nilpotent operators,
operator-valued polynomials

Received by editor(s):
February 19, 2010

Received by editor(s) in revised form:
August 16, 2010

Published electronically:
March 13, 2012

Additional Notes:
This research was partially supported by a grant from the National Science Foundation. Part of this research was done while the second author was at Texas A&M University, College Station

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.