Strict convexity of some subsets of Hankel operators

Authors:
Caixing Gu and Jonathan E. Shapiro

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2779-2789

MSC (2000):
Primary 47B35, 47B20

DOI:
https://doi.org/10.1090/S0002-9939-03-06873-4

Published electronically:
January 2, 2003

MathSciNet review:
1974335

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Abstract | References | Similar Articles | Additional Information

Abstract: We find some extreme points in the unit ball of the set of Hankel operators and show that the unit ball of the set of compact Hankel operators is strictly convex. We use this result to show that the collection of lower triangular Toeplitz contractions is strictly convex. We also find some extreme points in certain reduced Cowen sets and discuss cases in which they are or are not strictly convex.

**1.**V. M. Adamjan, D. Z. Arov, and M. G. Krein,*Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem*, Math USSR Sbornick**86**(1971) 34-75. MR**45:7505****2.**M. Cho, R. Curto, and W. Y. Lee, Triangular Toeplitz contractions and Cowen sets for analytic polynomials. Preprint, 2000.**3.**C.C. Cowen,*Hyponormal and subnormal Toeplitz operators*, Surveys of Some Recent Results in rational, Operator Theory, I (J.B. Conway and B.B. Morrel, eds.), Pitman Research Notes in Mathematics, Vol.**171**, Longman, 1988, pp. 155-167. MR**90j:47022****4.**C. C. Cowen,*Hyponormality of Toeplitz operators*, Proc. Amer. Math. Soc.**103**(1988), 809-812. MR**89f:47038****5.**C. Foias and A. Frazho,*The Commutant Lifting Approach to Interpolation Problems*, Birkhauser-Verlag, Boston, 1990. MR**92k:47033****6.**C. Foias, A. Tannenbaum, and G. Zames,*Sensitivity minimization for arbitrary SISO distributed plants*, Systems & Control Letters**8**(1987), 189-195. MR**88e:93045****7.**C. Gu and J. E. Shapiro,*Kernels of Hankel operators and hyponormality of Toeplitz operators*, Math. Ann.**319**(2001), 553-572. MR**2001m:47049****8.**B. Sz. Nagy and C. Foias,*Harmonic Analysis of Operators on Hilbert Space*, North-Holland, Amsterdam, 1970. MR**43:947****9.**Z. Nehari,*On bounded bilinear forms,*Ann. of Math.**65**(1957), 153-162. MR**18:633f****10.**N. Young,*An introduction to Hilbert space*, Cambridge University Press, Cambridge, UK, 1988. MR**90e:46001**

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Additional Information

**Caixing Gu**

Affiliation:
Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407

Email:
cgu@calpoly.edu

**Jonathan E. Shapiro**

Affiliation:
Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407

Email:
jshapiro@calpoly.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-06873-4

Keywords:
Hankel operator,
Toeplitz operator,
convex

Received by editor(s):
October 10, 2001

Received by editor(s) in revised form:
April 1, 2002

Published electronically:
January 2, 2003

Additional Notes:
The work of the first author was partially supported by the National Science Foundation Grant DMS-0075127 and both authors were supported by the SFSG Grants of California Polytechnic State University.

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
by the authors