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

Caixing Gu|Yucheng Li|Hexin Zhang
대한수학회 (The Korean Mathematical Society)|2024.07|대한수학회보 (Bulletin of the KMS)|제61권 제4호|pp. 949-957|DOI 10.4134/BKMS.b230504

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참고문헌

키워드

Dirichlet spaceMultiplication operatorVolterra operatorreducing subspace
주제분류수학

초록

Let $Y_{p}$ be the multiplication operator $M_{p}$ plus the Volterra operator $V_{p}$ induced by $p(z)$, where $p$ is a polynomial. Under a mild condition, we prove that $Y_{p}$ acting on the Dirichlet space $\mathfrak{D}$ is similar to multiplication operator $M_{p}$ acting on a subspace $S(\mathbb{D})$ of $\mathfrak{D}$. Furthermore, it shows that $T_{z^n}\,(n\geq2)$ has exactly $2^{n}$ reducing subspaces on $\mathfrak{D}$.

Abstract

Let $Y_{p}$ be the multiplication operator $M_{p}$ plus the Volterra operator $V_{p}$ induced by $p(z)$, where $p$ is a polynomial. Under a mild condition, we prove that $Y_{p}$ acting on the Dirichlet space $\mathfrak{D}$ is similar to multiplication operator $M_{p}$ acting on a subspace $S(\mathbb{D})$ of $\mathfrak{D}$. Furthermore, it shows that $T_{z^n}\,(n\geq2)$ has exactly $2^{n}$ reducing subspaces on $\mathfrak{D}$.

참고문헌 (20)

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