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 Algebra i Analiz, 2006, Volume 18, Issue 1, Pages 144–161 (Mi aa63)

Research Papers

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

S. Potta, E. Strouseb

a Department of Mathematics, University of Glasgow, Glasgow, UK
b Departement de Mathematiques Pures, Université Bordeaux I, Talence, France

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

Keywords: weighted Bergman spaces, Toeplitz operators, reproducing kernel thesis

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English version:
St. Petersburg Mathematical Journal, 2007, 18:1, 105–118

Bibliographic databases:

MSC: 47B35, 32A36
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Citation: S. Pott, E. Strouse, “Products of Toeplitz operators on the Bergman spaces $A_\alpha^2$”, Algebra i Analiz, 18:1 (2006), 144–161; St. Petersburg Math. J., 18:1 (2007), 105–118

Citation in format AMSBIB
\Bibitem{PotStr06} \by S.~Pott, E.~Strouse \paper Products of Toeplitz operators on the Bergman spaces~$A_\alpha^2$ \jour Algebra i Analiz \yr 2006 \vol 18 \issue 1 \pages 144--161 \mathnet{http://mi.mathnet.ru/aa63} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2225216} \zmath{https://zbmath.org/?q=an:1127.47028} \transl \jour St. Petersburg Math. J. \yr 2007 \vol 18 \issue 1 \pages 105--118 \crossref{https://doi.org/10.1090/S1061-0022-06-00945-9} 

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This publication is cited in the following articles:
1. Grudsky S., Vasilevski N., “On the structure of the $C^*$-algebra generated by Toeplitz operators with piece-wise continuous symbols”, Complex Anal. Oper. Theory, 2:4 (2008), 525–548
2. Lu Yu., Liu Ch., “Toeplitz and Hankel Products on Bergman Spaces of the Unit Ball”, Chin. Ann. Math. Ser. B, 30:3 (2009), 293–310
3. Kerr R., “Products of Toeplitz operators on a vector valued Bergman space”, Integral Equations Operator Theory, 66:3 (2010), 367–395
4. Mitkovski M., Wick B.D., “A Reproducing Kernel Thesis For Operators on Bergman-Type Function Spaces”, J. Funct. Anal., 267:7 (2014), 2028–2055
5. Michalska M., Sobolewski P., “Bounded Toeplitz and Hankel Products on the Weighted Bergman Spaces of the Unit Ball”, J. Aust. Math. Soc., 99:2 (2015), 237–249
6. Rahm R., “the Essential Norm of Operators on l(2)-Valued Bergman-Type Function Spaces”, Complex Anal. Oper. Theory, 10:1 (2016), 69–96
7. Sehba B.F., “On the Weighted Estimate of the Bergman Projection”, Czech. Math. J., 68:2 (2018), 497–511
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