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Products of Toeplitz operators on the harmonic Bergman space. (English) Zbl 1195.47014
The authors study Toeplitz operators with quasihomogeneous symbols acting on the harmonic Bergman space over the unit disk. The main result of the paper states that, given \(l_1,l_2 > 0\) and \(k_1, k_2 \in \mathbb{Z}\), the product \(T_{e^{ik_1\theta}r^{l_1}}T_{e^{ik_2\theta}r^{l_2}}\), where \(r =|z|\), is a Toeplitz operator if and only if \(k_1=k_2=0\), and in this case
\[ T_{r^{l_1}}T_{r^{l_2}}= \begin{cases}\frac{l_1}{l_1-l_2}T_{r^{l_1}} - \frac{l_2}{l_1-l_2}T_{r^{l_2}} &\text{if }l_1 \neq l_2,\\ T_{r^{l_1}}T_{r^{l_2}}=T_{r^{l_1}(1+ l_1\log r)}&\text{otherwise.}\end{cases} \]

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B38 Linear operators on function spaces (general)
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[1] Patrick Ahern and Željko Čučković, A theorem of Brown-Halmos type for Bergman space Toeplitz operators, J. Funct. Anal. 187 (2001), no. 1, 200 – 210. · Zbl 0996.47037
[2] Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89 – 102. · Zbl 0116.32501
[3] Boo Rim Choe and Young Joo Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999), no. 1, 163 – 174. · Zbl 0969.47023
[4] Željko učković and N. V. Rao, Mellin transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal. 154 (1998), no. 1, 195 – 214. · Zbl 0936.47015
[5] X.T. Dong and Z.H. Zhou, Algebraic properties of Toeplitz operators with separately quasi- homogeneous symbols on the Bergman space of the unit ball, J. Operator Theory, to appear. · Zbl 1234.47014
[6] Issam Louhichi, Elizabeth Strouse, and Lova Zakariasy, Products of Toeplitz operators on the Bergman space, Integral Equations Operator Theory 54 (2006), no. 4, 525 – 539. · Zbl 1109.47023
[7] Issam Louhichi and Lova Zakariasy, On Toeplitz operators with quasihomogeneous symbols, Arch. Math. (Basel) 85 (2005), no. 3, 248 – 257. · Zbl 1088.47019
[8] Reinhold Remmert, Classical topics in complex function theory, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. · Zbl 0895.30001
[9] Lova Zakariasy, The rank of Hankel operators on harmonic Bergman spaces, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1177 – 1180. · Zbl 1010.47018
[10] Ze-Hua Zhou and Xing-Tang Dong, Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball, Integral Equations Operator Theory 64 (2009), no. 1, 137 – 154. · Zbl 1195.47020
[11] Nina Zorboska, The Berezin transform and radial operators, Proc. Amer. Math. Soc. 131 (2003), no. 3, 793 – 800. · Zbl 1009.47015
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