Reducing subspaces of multiplication operators on the Dirichlet space

TL;DR

This paper investigates the structure of reducing subspaces for multiplication operators induced by finite Blaschke products on the Dirichlet space, revealing conditions for reducibility and equivalence to monomials.

Contribution

It characterizes when such operators are reducible on the Dirichlet space and identifies the form of Blaschke products that lead to reducibility.

Findings

Any two distinct nontrivial minimal reducing subspaces are orthogonal.

For orders 2 and 3, reducibility occurs iff the Blaschke product is equivalent to a monomial.

For order 4 and non-prime orders, reducibility can occur without the product being equivalent to a monomial.

Abstract

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $\phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_\phi$ are orthogonal. When the order $n$ of $\phi$ is $2$ or $3$, we show that $M_\phi$ is reducible on $D$ if and only if $\phi$ is equivalent to $z^n$. When the order of $\phi$ is $4$, we determine the reducing subspaces for $M_\phi$, and we see that in this case $M_\phi$ can be reducible on $D$ when $\phi$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $\phi$ is not a prime number. Furthermore, we show that $M_\phi$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $\phi = az^n$ for some unimodular constant $a$.