Contractions with polynomial characteristic functions I. Geometric approach
Authors:
Ciprian Foias and Jaydeb Sarkar
Journal:
Trans. Amer. Math. Soc. 364 (2012), 41274153
MSC (2010):
Primary 47A45, 47A20, 47A48, 47A56
Published electronically:
March 13, 2012
MathSciNet review:
2912448
Fulltext 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 quasisimilarity, is uniquely determined by . Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.
 1.
Bhaskar
Bagchi and Gadadhar
Misra, Constant characteristic functions and homogeneous
operators, J. Operator Theory 37 (1997), no. 1,
51–65. MR
1438200 (98a:47009)
 2.
Hari
Bercovici, Operator theory and arithmetic in
𝐻^{∞}, Mathematical Surveys and Monographs,
vol. 26, American Mathematical Society, Providence, RI, 1988. MR 954383
(90e:47001)
 3.
Douglas
N. Clark and Gadadhar
Misra, On homogeneous contractions and unitary representations of
𝑆𝑈(1,1), J. Operator Theory 30
(1993), no. 1, 109–122. MR 1302610
(96h:47015)
 4.
M.
J. Cowen and R.
G. Douglas, Complex geometry and operator theory, Acta Math.
141 (1978), no. 34, 187–261. MR 501368
(80f:47012), 10.1007/BF02545748
 5.
Ciprian
Foias and Arthur
E. Frazho, The commutant lifting approach to interpolation
problems, Operator Theory: Advances and Applications, vol. 44,
Birkhäuser Verlag, Basel, 1990. MR 1120546
(92k:47033)
 6.
I.
Gohberg, P.
Lancaster, and L.
Rodman, Matrix polynomials, Academic Press, Inc. [Harcourt
Brace Jovanovich, Publishers], New YorkLondon, 1982. Computer Science and
Applied Mathematics. MR 662418
(84c:15012)
 7.
Israel
Gohberg, Seymour
Goldberg, and Marinus
A. Kaashoek, Basic classes of linear operators,
Birkhäuser Verlag, Basel, 2003. MR 2015498
(2005g:47001)
 8.
I.
C. Gohberg and M.
G. Kreĭn, Introduction to the theory of linear
nonselfadjoint operators, Translated from the Russian by A. Feinstein.
Translations of Mathematical Monographs, Vol. 18, American Mathematical
Society, Providence, R.I., 1969. MR 0246142
(39 #7447)
 9.
Béla
Sz.Nagy and Ciprian
Foiaș, Harmonic analysis of operators on Hilbert space,
Translated from the French and revised, NorthHolland Publishing Co.,
AmsterdamLondon; American Elsevier Publishing Co., Inc., New York;
Akadémiai Kiadó, Budapest, 1970. MR 0275190
(43 #947)
 10.
Stephen
Parrott, On a quotient norm and the Sz.Nagy\thinspace\thinspace
Foiaş lifting theorem, J. Funct. Anal. 30
(1978), no. 3, 311–328. MR 518338
(81h:47006), 10.1016/00221236(78)900605
 11.
Vladimir
V. Peller, Hankel operators and their applications, Springer
Monographs in Mathematics, SpringerVerlag, New York, 2003. MR 1949210
(2004e:47040)
 12.
Joel
David Pincus, Commutators and systems of singular integral
equations. I, Acta Math. 121 (1968), 219–249.
MR
0240680 (39 #2026)
 13.
Leiba
Rodman, An introduction to operator polynomials, Operator
Theory: Advances and Applications, vol. 38, Birkhäuser Verlag,
Basel, 1989. MR
997092 (90k:47032)
 14.
Radu
I. Teodorescu, Fonctions caractéristiques constantes,
Acta Sci. Math. (Szeged) 38 (1976), no. 12,
183–185 (French). MR 0442721
(56 #1101)
 15.
Pei
Yuan Wu, Contractions with constant characteristic function are
reflexive, J. London Math. Soc. (2) 29 (1984),
no. 3, 533–544. MR 754939
(85k:47020), 10.1112/jlms/s229.3.533
 1.
 B. Bagchi and G. Misra, Constant characteristic functions and homogeneous operators, J. Operator Theory 37 (1997), no. 1, 5165. MR 98a:47009
 2.
 H. Bercovici, Operator theory and arithmetic in , Mathematical Surveys and Monographs, 26, American Mathematical Society, Providence, RI, 1988. MR 90e:47001
 3.
 D. N. Clark and G. Misra, On homogeneous contractions and unitary representations of SU, J. Operator Theory 30 (1993), no. 1, 109122. MR 96h:47015
 4.
 M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 34, 187261. MR 80f:47012
 5.
 C. Foias and A. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, 44. Birkhäuser Verlag, Basel, 1990. MR 92k:47033
 6.
 I. Gohberg, P. Lancaster and L. Rodman, Matrix polynomials, Academic Press, New York, 1982. MR 84c:15012
 7.
 I. C. Gohberg, S. Goldberg and M. Kaashoek, Basic classes of linear operators, Birkhäuser Verlag, Basel, 2003. MR 2005g:47001
 8.
 I. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, RI, 1969. MR 0246142 (39:7447)
 9.
 B. Sz.Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970. MR 0275190 (43:947)
 10.
 S. Parrott, On a quotient norm and the Sz.NagyFoias lifting theorem, J. Funct. Anal. 30 (1978), no. 3, 311328. MR 81h:47006
 11.
 V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics. SpringerVerlag, New York, 2003. MR 2004e:47040
 12.
 J. D. Pincus, Commutators and systems of singular integral equations, I, Acta Math. 121 (1968), 219249. MR 0240680 (39:2026)
 13.
 L. Rodman, An Introduction to Operator Polynomials, Operator Theory 38, Birkhäuser Verlag, Basel and Boston, 1989. MR 90k:47032
 14.
 R. Teodorescu, Fonctions caractéristiques constantes, Acta Sci. Math. (Szeged) 38 (1976), no. 12, 183185. MR 0442721
 15.
 P.Y. Wu, Contractions with constant characteristic function are reflexive, J. London Math. Soc. (2) 29 (1984), no. 3, 533544. MR 0754939
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
47A45,
47A20,
47A48,
47A56
Retrieve articles in all journals
with MSC (2010):
47A45,
47A20,
47A48,
47A56
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, StatMath Unit, 8th Mile, Mysore Road, RVCE Post, Bangalore 560059, Karnataka, India
Email:
jay@isibang.ac.in, jaydeb@gmail.com
DOI:
http://dx.doi.org/10.1090/S00029947201205450X
Keywords:
Characteristic function,
model,
weighted shifts,
nilpotent operators,
operatorvalued 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.
