Bulletin of the
Korean Mathematical Society

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Online first article May 16, 2024

Copyright © The Korean Mathematical Society.

On similarity and reducing subspaces of a class of operator on the Dirichlet space

Caixing Gu, Yucheng Li, and Hexin Zhang

California Polytechnic State University, Hebei Normal University


Let $T_{p}$ be the multiplication operator $M_{p}$ plus the Volterra operator $V_{p}$ induced by $p(z)$, where $p(z)=\sum\limits_{k=0}^{n}m_k(z)$, and $m_k(z)=d_kz^k, d_k\in\mathbb{C}$. Under a mild condition, we prove that $T_{p}$ acting on the Dirichlet space $\mathfrak{D}$ is similar to multiplication operator $M_{p}$ acting on $S(\mathbb{D})$. Furthermore, it shows that $T_{m_n}\,(n\geq2)$ has exactly $2^{n}$ reducing subspaces on $\mathfrak{D}$.

Keywords: multiplication operator, Volterra operator, reducing subspace, Dirichlet space.

MSC numbers: 47B35, 32A36

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