FINITE RANK COMMUTATORS AND SEMICOMMUTATORS OF QUASIHOMOGENEOUS TOEPLITZ OPERATORS ˘ ZELJKO ˘ CU ˘ CKOVI ´ C AND ISSAM LOUHICHI Abstract. We study finite rank semicommutators and commutators of Toeplitz operators on the Bergman space with quasihomogeneous symbols. We show that in this context, the situation is different from the case of harmonic Toeplitz operators. 1. Introduction Let D be the unit disc in the complex plane C, and dA = rdr dθ π be the normalized Lebesgue area measure so that the measure of D equals 1. The Bergman space L 2 a is the Hilbert space consisting of the analytic functions on D that are also square integrable with respect to the measure dA. We denote the inner product in L 2 (D, dA) by <, >. It is well known that L 2 a is a closed subspace of the Hilbert space L 2 (D, dA), and has the set { √ n +1z n | n ≥ 0} as an orthonormal basis. We let P be the orthogonal projection from L 2 (D, dA) onto L 2 a . For a bounded function φ on D, the Toeplitz operator T φ with symbol φ is defined by T φ (h)= P (φh) for h ∈ L 2 a . In the last two decades, a lot of work has been done in understanding the alge- braic properties of Toeplitz operators on the Bergman space. This includes studying the semicommutators and commutators of Toeplitz operators. For two Toeplitz op- erators T φ and T ψ we define the semicommutator and the commutator respectively by (T φ ,T ψ ]= T φψ - T φ T ψ and [T φ ,T ψ ]= T φ T ψ - T ψ T φ . Commuting Toeplitz operators with harmonic symbols were characterized by Axler and the first author [4], and essentially commuting by Stroethoff [16]. The zero semicommutators were studied by Zheng [18]. The natural question to ask is when these objects are compact or finite rank. In this respect we should mention here the characterization of compact Toeplitz operators by Axler and Zheng [5] in terms of the Berezin transform of the symbol. In this context we also mention the work of Su´arez [17]. Very recently, Luecking [14] has proved that the only finite rank Toeplitz operator is the zero one. Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols were characterized by Guo, Sun and Zheng [10]. About the same time, finite rank perturbations of related products of Date : July 7, 2007. 2000 Mathematics Subject Classification. Primary 47B35; Secondary 47L80. Key words and phrases. Toeplitz operators, semicommutators, commutators, Mellin transform, finite rank operators. 1