Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk

Abstract

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.

Multiplication operators on the Bergman space via the Hardy space of the bidisk

“…Another one is that we have obtained a complete description of nontrivial minimal reducing subspaces of the multiplication operator by φ on the Bergman space of the unit disk for the fourth order Blaschke product φ [31].…”

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

“…In 2009, Guo et al [7] proved that if is a Blaschke product of degree 3, then the number of minimal reducing subspaces of is at most 3. For finite Blaschke product , they also [8][9][10]. Finally, an affirmative answer to the conjecture is given by Douglas et al [11].…”

Totally Abelian Toeplitz operators and geometric invariants associated with their symbol curves

“…Below, H denotes the Hardy space H 2 (D) or the Bergman space L 2 a (D). As presented below is the remarkable theorem on commutants for analytic Toeplitz operators, due to Thomson and Cowen [21,22,5]; also see [11,Chapter 3] for a detailed discussion and see [7,19,13] for related work on this line. One defines a quantity b(φ) to be the maximum of orders of B, for which there is a function ψ in H ∞ (D) such that φ = ψ( B), and b(φ) is called Cowen-Thomson order of φ.…”