Reducing subspaces of multiplication operators with the symbol αz k + βw l on $$L_a^2 (\mathbb{D}^2 )$$

Abstract

Multiplication operators defined on function spaces have been receiving enormous attention from both operator-theoretic and function-theoretic experts. One of the problems is to study reducing subspaces of them. The one-variable case has obtained fruitful remarkable results. However, little has been done in the multi-variable case. Under the setting of the Bergman space \(L_a^2 (\mathbb{D}^2 )\), this paper addresses those multiplication operators M p defined by special polynomials p, where p(z,w) = αz k +βw l , α, β ∈ ℂ. Those reducing subspaces of M p are completely determined.