The Commuting Of Toeplitz Operators With Quasihomogeneous Symbols On The Bergman Space

As we all know, the commuting of Toeplitz operators with bounded symbols on the Bergman space is still an open question. The question has been partially studied by many researchers, among all the results the most well-known is the characterization of commuting Toeplitz operators with harmonic symbols on the Bergman space.In1998, N. Rao defined the concept of quasihomogeneous function for the first time in his paper, which gives us a new way to study the commuting of Toeplitz operators. Based on N. Rao’s results, I. Louhichi talked about the commutativity of Toeplitz operators with quasihomo-geneous symbols.In this paper, we generalize the results of I. Louhichi to some extents. Suppose eφipθ, eψisθ are two quasihomogeneous functions, where p, s are positive integers and φ,ψ are bounded radial functions, we study the commuting of Teφipθ and Teψisθ. Under the assumpution of that the Mellin transform of one of the radial functions φ has a zero point, we get our results:two quasihomogeneous Toeplitz operators with positive degree commute only in certain trivial case and for any∫L∞(D, dA) Teφipθ commutes with Tf if and only if f=C1eφipθ+C2, where C1and C2are constants.Meanwhile, we characterize the commutativity of Toeplitz operators with symbols of finite sum of quasihogeneous functions. We show that one Toeplitz operator with symbol of finite sum of positive degree qusihomogeneous function commutes with another such Toeplitz operator of negative degree only in a trivial case.