Journal of Operator Theory

Volume 66, Issue 1, Summer 2011 pp. 193-207.

Algebraic properties of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the unit ball

Authors: Xing-Tang Dong (1) and Ze-Hua Zhou (2)
Author institution: (1) Department of Mathematics, Tianjin University, Ti-anjin, 300072, P.R. China
(2) Department of Mathematics, Tianjin University, Ti-anjin, 300072, P.R. China

Summary: In this paper we discuss some algebraic properties of Toeplitz operators with separately quasihomogeneous symbols (i.e., symbols being of the form $\xi^{k}\varphi(|z_1|,\ldots,|z_n|)$) on the Bergman space of the unit ball in $\mathbb{C}^n$. We provide a decomposition of $L^{2}(B_{n},\mathrm dv)$, then we use it to show that the zero product of two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Also, we describe the commutant of a Toeplitz operator whose symbol is radial.

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