## 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

## 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.## References

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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
- 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: https://doi.org/10.1090/S0002-9947-2012-05450-X
- MathSciNet review: 2912448