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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
Published electronically: March 13, 2012
MathSciNet review: 2912448
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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

$\displaystyle T = \begin {bmatrix}S & * & *\\ 0 & N & *\\ 0& 0& C \end{bmatrix},$    

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

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.