# Article

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Keywords:
quasihomogeneous Toeplitz operator; Mellin transform
Summary:
We present sufficient conditions for the existence of $p$th powers of a quasihomogeneous Toeplitz operator $T_{{\rm e}^{{\rm i} s\theta }\psi }$, where $\psi$ is a radial polynomial function and $p$, $s$ are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.
References:
[1] Ahern, P., Čučković, Ž.: A theorem of Brown-Halmos type for Bergman space Toeplitz operators. J. Funct. Anal. 187 (2001), 200-210. DOI 10.1006/jfan.2001.3811 | MR 1867348 | Zbl 0996.47037
[2] Cohen, A. M.: Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms 5. Springer, New York (2007). DOI 10.1007/978-0-387-68855-8 | MR 2325479 | Zbl 1127.65094
[3] Čučković, Ž., Rao, N. V.: Mellin transform, monomial symbols, and commuting Toeplitz operators. J. Funct. Anal. 154 (1998), 195-214. DOI 10.1006/jfan.1997.3204 | MR 1616532 | Zbl 0936.47015
[4] Godement, R.: Analysis III. Analytic and Differential Functions, Manifolds and Riemann Surfaces. Universitext. Springer, Cham (2015). DOI 10.1007/978-3-319-16053-5 | MR 3328588 | Zbl 1318.30001
[5] Louhichi, I.: Powers and roots of Toeplitz operators. Proc. Am. Math. Soc. 135 (2007), 1465-1475. DOI 10.1090/S0002-9939-06-08626-6 | MR 2276656 | Zbl 1112.47023
[6] Louhichi, I., Rao, N. V.: Roots of Toeplitz operators on the Bergman space. Pac. J. Math. 252 (2011), 127-144. DOI 10.2140/pjm.2011.252.127 | MR 2862145 | Zbl 1237.47033
[7] Louhichi, I., Rao, N. V., Yousef, A.: Two questions on products of Toeplitz operators on the Bergman space. Complex Anal. Oper. Theory 3 (2009), 881-889. DOI 10.1007/s11785-008-0097-3 | MR 2570117 | Zbl 1195.47018
[8] Louhichi, I., Strouse, E., Zakariasy, L.: Products of Toeplitz operators on the Bergman space. Integral Equations Oper. Theory 54 (2006), 525-539. DOI 10.1007/s00020-005-1369-1 | MR 2222982 | Zbl 1109.47023
[9] Louhichi, I., Zakariasy, L.: On Toeplitz operators with quasihomogeneous symbols. Arch. Math. 85 (2005), 248-257. DOI 10.1007/s00013-005-1198-0 | MR 2172383 | Zbl 1088.47019
[10] Remmert, R.: Classical Topics in Complex Function Theory. Graduate Texts in Mathematics 172. Springer, New York (1998). DOI 10.1007/978-1-4757-2956-6 | MR 1483074 | Zbl 0895.30001

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