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

Full-text PDF

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

where and are unilateral shifts of arbitrary multiplicities and is nilpotent. We prove that the dimension of ker and the dimension of are unitary invariants of and that , up to a quasi-similarity, is uniquely determined by . 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

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

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

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.