Algebraic Properties of Toeplitz Operators on the Polydisk

and Applied Analysis 3 For commuting problem, in 1963, Brown and Halmos 2 showed that two bounded Toeplitz operators Tφ and Tψ on the classical Hardy space commute if and only if i both φ and ψ are analytic, ii both φ and ψ are analytic, or iii one is a linear function of the other. On the Bergman space of the unit disk, some similar results were obtained for Toeplitz operators with bounded harmonic symbols or analytic symbols see 2–4 . The problem of characterizing commuting Toeplitz operators with arbitrary bounded symbols seems quite challenging and is not fully understood until now. In recent years, by Mellin transform, some results with quasihomogeneous symbols it is of the form eφ, where φ is a radial function or monomial symbols were obtained see 5–7 . On the Hardy and Bergman spaces of several complex variables, the situation is much more complicated. On the unit ball, Toeplitz operators with pluriharmonic or quasihomogeneous symbols were studied in 1, 8–11 . On the polydisk, some results about Toeplitz operators with pluriharmonic symbols were obtained in 10, 12–14 . For finite-rank product problem, Luecking recently proved that a Toeplitz operator with measure symbol on the Bergman space of unit disk has finite rank if and only if its symbols are a linear combination of point masses see 15 . In 16 , Choe extended Luecking’s theorem to higher-dimensional cases. Using those results, Le studied finite-rank products of Toeplitz operators on the Bergman space of the unit disk and unit ball in 17, 18 . Motivated by recent work in 1, 5, 7, 17, 18 , we define quasihomogeneous functions on the polydisk and study Toeplitz operators with quasihomogeneous symbols on the Bergman space of the polydisk. The present paper is assembled as follows. In Section 2, we introduce Mellin transform, Toeplitz operators with quasihomogeneous symbols and property P . In Section 3, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. In Sections 4 and 5, we prove that finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols must be zero operator and we also solve the finite-rank product problem for Toeplitz operators on the Bergman space of the polydisk. 2. Mellin Transform, Toeplitz Operators with Quasihomogeneous Symbols and Property (P) For any multi-index α α1, . . . , αn ∈ N here N denotes the set of all nonnegative integers , we write aα α1 · · ·αn and z z1 1 · · · zn n for z z1, . . . , zn ∈ D. The standard orthonormal basis for A2 is {eα : α ∈ Nn}, where eα z √ α1 1 · · · αn 1 z, α ∈ N, z ∈ D. 2.1 For two n-tuples of integers α α1, . . . , αn and β β1, . . . , βn , we define α β if αj > βj for all 0 ≤ j ≤ n. Similarly, we write α β if αj ≥ βj for all 1 ≤ j ≤ n and α β if otherwise. We also define α ⊥ β if α1β1 · · · αnβn 0 and α − β α1 − β1, . . . , αn − βn . For any k k1, . . . , kn ∈ Z, particularly we write k1 k1, . . . , k1 and put k∗ |k1|, . . . , |kn| , k 1/2 k∗ k and k− 1/2 k∗ − k . Then, k , k− 0, k k − k−, and k ⊥ k−. 4 Abstract and Applied Analysis Recall that a function φ on D is radial if and only if φ z depends only on |z1|, |z2|, . . . , |zn| , that is, φ e1z1, e2z2, . . . , enzn φ z1, z2, . . . , zn for any θ1, θ2, . . . , θn ∈ R. For any function f ∈ L1 D, dV , we define the radicalization of f by rad ( f ) z1, z2, . . . , zn 1 2π n ∫2π