Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

Abstract EN

We consider the reducing subspaces of M z N on A α 2 ( D k ) , where k ≥ 3 , z N = z 1 N 1 ⋯ z k N k , and N i ≠ N j for i ≠ j .

We prove that each reducing subspace of M z N is a direct sum of some minimal reducing subspaces.

We also characterize the minimal reducing subspaces in the cases that α = 0 and α ∈ ( - 1 , + ∞ ) ∖ Q , respectively.

Finally, we give a complete description of minimal reducing subspaces of M z N on A α 2 ( D 3 ) with α > - 1 .