Geometric constructions of thin Blaschke products and reducing subspace problem

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 $M_B^{\ast }$. 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 $\varphi$, $M_{\varphi }$ 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.