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Available in print format 		St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(e) ISSN 1061-0022(p)

     

Products of Toeplitz operators on the Bergman spaces $ A_\alpha^2$

Author(s): S. Pott; E. Strouse
Original publication: Algebra i Analiz, tom 18 (2006), nomer 1.
Journal: St. Petersburg Math. J. 18 (2007), 105-118.
MSC (2000): Primary 47B35, 32A36
Posted: November 27, 2006
MathSciNet review: 2225216
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Abstract | References | Similar articles | Additional information

Abstract: We give a sufficient and a necessary condition for the product of Toeplitz operators $ T^\alpha_f T^\alpha_{\bar g}$, with $ f,g$ analytic, to be bounded on the weighted Bergman space $ L^2_a(\mathbb{D},(1-\vert z\vert^2)^\alpha dA)$. We also show that the only compact product of weighted Toeplitz operators is the trivial one.

References:

[HaNik]
V. P. Havin and N. K. Nikolski, Stanislav Aleksandrovich Vinogradov, his life and mathematics, Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 1-18. MR 1771747 (2001b:01025a)

[HKZ]
H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. MR 1758653 (2001c:46043)

[Lu]
D. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), 85-111. MR 0778090 (86g:30002)

[LP]
N. Lusin and J. Priwaloff, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. (3) 42 (1925), no. 3, 143-191.

[M]
Mathworld Website http://mathworld.wolfram.com/BetaFunction.html

[N1]
F. Nazarov, A counterexample to Sarason's conjecture, Preprint, www.math.msu.edu/~fedja/pepr.html

[N2]
F. Nazarov, Private communication.

[S]
D. Sarason, Products of Toeplitz operators, Linear and Complex Analysis. Problem Book 3, Part I (V. P. Havin and N. K. Nikolski, eds.), Lecture Notes in Math., vol. 1573, Springer-Verlag, Berlin, 1994.

[Sm]
M. Smith, Preprint, 2004.

[StZh1]
K. Stroethoff and D. Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), 289-313. MR 1726756 (2000i:47053)

[StZh2]
-, Invertible Toeplitz products, J. Funct. Anal. 195 (2002), no. 1, 48-70. MR 1934352 (2003g:47051)

[TVZh]
S. Treil, A. Volberg, and D. Zheng, Hilbert transform, Toeplitz operators and Hankel operators, and invariant $ A\sb \infty$ weights, Rev. Mat. Iberoamericana 13 (1997), no. 2, 319-360. MR 1617653 (2000e:47061)

[VMü]
V. Müller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), no. 4, 979-989. MR 1112498 (93e:47016)

[Zh]
D. Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators, J. Funct. Anal. 138 (1996), no. 2, 477-501. MR 1395967 (97e:47040)

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Additional Information:

S. Pott
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom
Email: sp@maths.gla.ac.uk

E. Strouse
Affiliation: Departement de Mathématiques Pures, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence Cedex, France
Email: Elizabeth.Strouse@math.u-bordeaux1.fr

DOI: 10.1090/S1061-0022-06-00945-9
PII: S 1061-0022(06)00945-9
Keywords: Weighted Bergman spaces, Toeplitz operators, reproducing kernel thesis
Received by editor(s): 3/OCT/2005
Posted: November 27, 2006
Additional Notes: This work was supported by the European Network on Analysis and Operators (HPRN CT 2000 00116), by a grant by the Nuffield Foundation, and by EPSRC. The first author would like to thank the Departement de Mathématiques Pures, Université Bordeaux I, for their hospitality
Copyright of article: Copyright 2006, American Mathematical Society




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