Multiplication Operators and M-Berezin Transforms

Lattices of reducing subspaces of multiplication operators acting on the Bergman space induced by finite Blaschke products are studied. A complete description of the lattices of reducing subspaces of multiplication operators induced by Blaschke products of order three or order four is given. It is proved that, for the multiplication operator acting on the Bergman space induced by Blaschke product of order three or order four, the number of minimal reducing subspaces equals the number of connected components of the Riemann surface associated to the composition of the inverse of the Blaschke product and the Blaschke product itself. A characterization about the compactness of certain operators in the Toeplitz algebra acting on Bergman spaces of several complex variables is obtained via m-Berezin transforms.