Reducing subspaces for a class of multiplication operators on the Dirichlet space

Zhao, Liankuo

doi:10.1090/S0002-9939-09-09859-1

Reducing subspaces for a class of multiplication operators on the Dirichlet space
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Proc. Amer. Math. Soc. 137 (2009), 3091-3097 Request permission

Abstract:

In this paper, we discuss reducing subspaces of multiplication operators $M_\phi$ on the Dirichlet space $\mathcal {D}$ defined by a Blaschke product $\phi$ with two zeros $a$, $b$ in the unit disk $\mathbb {D}$ and show that when $a+b=0$, $M_\phi$ has two proper ones; otherwise it has none. This is different from the cases of the Hardy space and the Bergman space.

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