Invariant subspaces of Toeplitz operators and uniform algebras (Q2481524)

The author studies the invariant subspaces of Toeplitz operators in an abstract Hardy space \(H^p\). Let \(\operatorname{Lat} T_{\varphi}\) be the set of all invariant subspaces of a Toeplitz operator \(T_{\varphi}\) and \(\operatorname{Lat} \mathcal{A}=\cap \{\operatorname{Lat} T_{\varphi}:\varphi \in H^{\infty}\}\), where \(\mathcal{A}=\{T_{\varphi}:\varphi\in H^{\infty}\}\). In particular, the following four questions are considered in the abstract setting. (1) \(\operatorname{Lat} T_{\varphi} \supseteq \operatorname{Lat}\mathcal{A} \Longrightarrow T_{\varphi} \in \mathcal{A} \,?\) (2) \(\operatorname{Lat}T_{\varphi} \subsetneq \operatorname{Lat}\mathcal{A} \Longrightarrow \operatorname{Lat}T_{\varphi} =\{\langle0\rangle, H^2\} \,?\) (3) \(\operatorname{Lat} \mathcal{A}^* \cap \operatorname{Lat}\mathcal{A}=\{\langle0\rangle, H^2\}\), where \(\mathcal{A^*}=\{T_{\varphi}^*:\varphi\in H^{\infty}\} \,?\) (4) What is \(\operatorname{Lat} T_{\varphi} \cap \operatorname{Lat}\mathcal{A}\)\,?