Geometric constructions of thin Blaschke products and reducing subspace problem

Abstract

In this paper, we mainly study geometric constructions of thin Blaschke products B and reducing subspace problem of multiplication operators induced by such symbols B on the Bergman space. Considering such multiplication operators M B , we present a representation of those operators commuting with both M B and M * B . It is shown that for "most" thin Blaschke products B, M B is irreducible, i.e. M B has no nontrivial reducing subspace; and such a thin Blaschke product B is constructed. As an application of the methods, it is proved that for "most" finite Blaschke products φ, M φ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for when M B defined by a thin Blaschke product B has a nontrivial reducing subspace.

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

“…Finally, an affirmative answer to the conjecture is given by Douglas et al [11]. Furthermore, when is an infinite Blaschke product, some relative results are obtained by Guo and Huang in [12,13].…”

Multiplication Operators on the Bergman Space

“…The following is from [GH4]. By Theorem 2.3.1, such an analytic continuation is unique, and one can show that this Q is necessarily a local inverse of B.…”

“…Proof The following proof is from [GH4]. Suppose B is a thin Blaschke product with its zero sequence fz n g. By Proposition 2.1.…”

“…Furthermore, Guo and Huang [GH4] obtained the following. Furthermore, Guo and Huang [GH4] obtained the following.…”

“…First we need a lemma [GH4] of independent interest. First we need a lemma [GH4] of independent interest.…”