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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Contractions with polynomial characteristic functions I. Geometric approach
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by Ciprian Foias and Jaydeb Sarkar PDF
Trans. Amer. Math. Soc. 364 (2012), 4127-4153 Request permission


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:,
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4127-4153
  • MSC (2010): Primary 47A45, 47A20, 47A48, 47A56
  • DOI:
  • MathSciNet review: 2912448