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## 2021年广州大学算子理论与算子代数学术交流会

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via composition operators. The result of Nordgren,Rossenthal and Wintrobe shown that ,for a

hyperbolic automorphism $\varphi$ of the unit disk $\mathbb{D}$, the ISP has a positive solution if and only if every minimal non-trivial invariant subspace of $C_{\varphi}$ on $H^{2}(\mathbb{D})$ is one dimensional. Recently, Carm and Noor complete the old solution and generalize it to the case for general linear fractional self-map of $\mathbb{D}$. Thus we can attack the ISP by studying a simple composition operator $C_{\varphi_{a}}$ on the Hardy space $H^{2}(\mathbb{D})$, where $\varphi_{a}(z)=az+(1-a)$ with $a\in (0,1)$.