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].

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

ABSTRACT. m-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the m-Berezin transform as a linear operator from the space of bounded operators to L ∞ is found. We show that the m-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the mBerezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

In this paper, an equivalent condition for collectively compact Hankel operator sequences is given. Our result extends the well-known characterization of single compact Hankel operators due to Hartman. ᮊ 1997 Academic Press