In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in C n . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form ξ k ϕ is studied, where k ∈ Z n and ϕ is a radial function. (2000). Primary 47B35; Secondary 32A36. Mathematics Subject Classification

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We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators T f Tg on the harmonic Bergman space is equal to a Toeplitz operator T h , then the product TgT f is also the Toeplitz operator T h , and hence T f commutes with Tg. From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator. D f (w)K z (w) dA(w)

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In this paper, we first discuss some basic results concerning Toeplitz operators with quasihomogeneous symbols (i.e., symbols being of the form e ipθ ϕ, where ϕ is a radial function) on the harmonic Bergman space. Then we determine when the product of two Toeplitz operators with quasihomogeneous symbols is a Toeplitz operator. 1 (1−wz) 2 be the well-known reproducing kernel for the analytic Bergman space L 2 a consisting of all L 2 -analytic functions on D. The well-known Bergman projection P is then the integral operator P f(z) = D f (w)K z (w)dA(w)