Reducibility and unitary equivalence for a class of multiplication operators on the Dirichlet space
Volume 220 / 2014
Studia Mathematica 220 (2014), 141-156
MSC: Primary 47B35; Secondary 32A36.
DOI: 10.4064/sm220-2-3
Abstract
We consider the reducibility and unitary equivalence of multiplication operators on the Dirichlet space. We first characterize reducibility of a multiplication operator induced by a finite Blaschke product and, as an application, we show that a multiplication operator induced by a Blaschke product with two zeros is reducible only in an obvious case. Also, we prove that a multiplication operator induced by a multiplier $\phi $ is unitarily equivalent to a weighted shift of multiplicity 2 if and only if $\phi =\lambda z^2$ for some unimodular constant $\lambda $.
