Geometric constructions of thin Blaschke products and reducing subspace problem

TL;DR

This paper explores geometric methods to analyze thin Blaschke products and their associated multiplication operators on the Bergman space, revealing conditions for irreducibility and characterizing reducing subspaces.

Contribution

It introduces a geometric construction approach for thin Blaschke products and characterizes when their multiplication operators have nontrivial reducing subspaces.

Findings

Most thin Blaschke product multiplication operators are irreducible.

Most finite Blaschke product operators have exactly two minimal reducing subspaces.

A geometric criterion for the existence of nontrivial reducing subspaces is established.

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 $\phi$, $M_\phi$ 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.