Triangular Toeplitz contractions and Cowen sets for analytic polynomials
Authors:
Muneo Cho, Raúl E. Curto and Woo Young Lee
Journal:
Proc. Amer. Math. Soc. 130 (2002), 35973604
MSC (2000):
Primary 47B35, 15A57, 15A60; Secondary 47B20, 30D50
Published electronically:
May 8, 2002
MathSciNet review:
1920039
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be the collection of lower triangular Toeplitz matrices and let be the collection of lower triangular Toeplitz contractions. We show that is compact and strictly convex, in the spectral norm, with respect to ; that is, is compact, convex and , where and denote the topological boundary with respect to and the set of extreme points, respectively. As an application, we show that the reduced Cowen set for an analytic polynomial is strictly convex; more precisely, if is an analytic polynomial and if , then is strictly convex. This answers a question of C. Cowen for the case of analytic polynomials.
 [Con]
J.B. Conway, A Course in Functional Analysis, SpringerVerlag, New York, 1985. MR 86h:46001
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C.C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of Some Recent Results in Operator Theory, I (J.B. Conway and B.B. Morrel, eds.), Pitman Research Notes in Mathematics, Vol 171, Longman, 1988, pp. 155167. MR 90j:47022
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C.C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), 809812. MR 89f:47038
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K.R. Davidson, Nest Algebras, Pitman Res. Notes Math. Ser., Vol 191, Longman, 1988. MR 90f:47062
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K. deLeeuw and W. Rudin, Extreme points and extremal problems in , Pacific J. Math. 8 (1958), 467485. MR 20:5426
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D.R. Farenick and W.Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996), 41534174. MR 97k:47027
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J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
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I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol II, Birkhäuser Verlag, Basel, 1993. MR 95a:47001
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C.R. Johnson and L. Rodman, Completion of Toeplitz partial contractions, SIAM J. Matrix Anal. Appl. 9 (1988), 159167. MR 89f:47040
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P.R. Halmos, A Hilbert Space Problem Book, SpringerVerlag, New York, 1982. MR 84e:47001
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T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), 753767. MR 93j:47040
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G. Nævdal, On the completion of partially given triangular Toeplitz matrices to contractions, SIAM J. Matrix Anal. Appl. 14 (1993), 545552. MR 94a:47025
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I. Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt, J. Reine Angew. Math. 147 (1917), 205232.
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S. Takahashi, Extension of the theorems of CarathéodoryToeplitzSchur and Pick, Pacific J. Math. 138 (1989), 391399. MR 90d:30105
 [Zhu]
K. Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equations Operator Theory 21 (1995), 376381. MR 95m:47044
 [Con]
 J.B. Conway, A Course in Functional Analysis, SpringerVerlag, New York, 1985. MR 86h:46001
 [Cow1]
 C.C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of Some Recent Results in Operator Theory, I (J.B. Conway and B.B. Morrel, eds.), Pitman Research Notes in Mathematics, Vol 171, Longman, 1988, pp. 155167. MR 90j:47022
 [Cow2]
 C.C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), 809812. MR 89f:47038
 [Da]
 K.R. Davidson, Nest Algebras, Pitman Res. Notes Math. Ser., Vol 191, Longman, 1988. MR 90f:47062
 [dLR]
 K. deLeeuw and W. Rudin, Extreme points and extremal problems in , Pacific J. Math. 8 (1958), 467485. MR 20:5426
 [FL]
 D.R. Farenick and W.Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996), 41534174. MR 97k:47027
 [Ga]
 J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
 [GGK]
 I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol II, Birkhäuser Verlag, Basel, 1993. MR 95a:47001
 [JR]
 C.R. Johnson and L. Rodman, Completion of Toeplitz partial contractions, SIAM J. Matrix Anal. Appl. 9 (1988), 159167. MR 89f:47040
 [Ha]
 P.R. Halmos, A Hilbert Space Problem Book, SpringerVerlag, New York, 1982. MR 84e:47001
 [NT]
 T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), 753767. MR 93j:47040
 [Næ]
 G. Nævdal, On the completion of partially given triangular Toeplitz matrices to contractions, SIAM J. Matrix Anal. Appl. 14 (1993), 545552. MR 94a:47025
 [Sch]
 I. Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt, J. Reine Angew. Math. 147 (1917), 205232.
 [Ta]
 S. Takahashi, Extension of the theorems of CarathéodoryToeplitzSchur and Pick, Pacific J. Math. 138 (1989), 391399. MR 90d:30105
 [Zhu]
 K. Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equations Operator Theory 21 (1995), 376381. MR 95m:47044
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Additional Information
Muneo Cho
Affiliation:
Department of Mathematics, Kanagawa University, Yokohama 2218686, Japan
Email:
chiyom01@kanagawau.ac.jp
Raúl E. Curto
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email:
curto@math.uiowa.edu
Woo Young Lee
Affiliation:
Department of Mathematics, SungKyunKwan University, Suwon 440746, Korea
Email:
wylee@yurim.skku.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002993902066285
PII:
S 00029939(02)066285
Keywords:
Triangular Toeplitz contractions,
hyponormal Toeplitz operators
Received by editor(s):
September 7, 2000
Received by editor(s) in revised form:
July 2, 2001
Published electronically:
May 8, 2002
Additional Notes:
The second author’s work was partially supported by NSF research grant DMS9800931
The third author’s work was partially supported by KOSEF research project No. R01200000003
Communicated by:
David R. Larson
Article copyright:
© Copyright 2002 American Mathematical Society
