Toeplitz Operators and Hankel Operators on the Hardy Space of the Unit Sphere

The object of this present paper is to study Toeplitz operators and Hankel operators on the Hardy space of the unit sphere S in C through the generalized area integral of harmonic functions on the unit ball B in C. In particular we consider the question of when the product of two Toeplitz operators is a compact perturbation of a Toeplitz operator. It follows from a theorem in [DJ] that T,T can be a compact perturbation of a Toeplitz operator only when it is a compact perturbation of T, . As is well known, the condition that either , or is in H implies that T,T =T . On the unit circle, Brown and Halmos [BH] showed that T,T =T, exactly when either , or is in H . But it is not known whether T,T =T, implies that either , or is in H when n is greater than 1. On the unit circle, Axler, Chang, and Sarason [ACS] found a sufficient condition, which is in terms of Douglas algebras, for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator. Later Volberg [V] proved that their condition is also necessary. Recently we [Z] have obtained an elementary necessary and sufficient condition for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator on the unit circle. In higher dimensions the theory of function algebras is so complicated, and it is not even known that the Carleson corona theorem holds for the unit ball. Also there are not so article no. FU973110