In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on ∂D. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz operators, compact Hankel operators, and appropriate products of these operators.

Abstract. We give a new and simple compactness criterion for composition operators C ϕ on BMOA and the Bloch space in terms of the norms of ϕ n in the respective spaces.

In this paper, using the group-like property of local inverses of a finite Blaschke product φ, we will show that the largest C * -algebra in the commutant of the multiplication operator M φ by φ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of φ −1 • φ over the unit disk. If the order of the Blaschke product φ is less than or equal to eight, then every C * -algebra contained in the commutant of M φ is abelian and hence the number of minimal reducing subspaces of M φ equals the number of connected components of the Riemann surface of φ −1 • φ over the unit disk.1991 Mathematics Subject Classification. 47B35, 30D50, 46E20.

In this paper we study Hankel operators and Toeplitz operators through a distribution function inequality on the Lusin area integral function and the Littlewood Paley theory. A sufficient condition and a necessary condition are obtained for the boundedness of the product of two Hankel operators. They lead to a way to approach Sarason's conjecture on products of Toeplitz operators and shed light on the compactness of the product of Hankel operators. An elementary necessary and sufficient condition for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator is obtained. Moreover, a necessary condition is given for the product of Hankel operators to be in the commutator ideal of the algebra generated by the Toeplitz operators with symbols in a Sarason algebra.

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Abstract. In this paper, we develop a machinery to study multiplication operators on the Bergman space via the Hardy space of the bidisk. Using the machinery we study the structure of reducing subspaces of a multiplication operator on the Bergman space. As a consequence, we completely classify reducing subspaces of the multiplication operator by a Blaschke product φ with order three on the Bergman space to solve a conjecture of Zhu [40].

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