Bounded Symbols and One-Component Inner Functions

Let $θ$ be a nonconstant inner function, we prove that every bounded truncated Toeplitz operator on the model space $K_θ=H^2\ominusθH^2$ admits a bounded symbol if and only if $θ$ is one-component, that is, if $\{z\in\mathbb D:|θ(z)|<r\}$ is nonempty and connected for some $r\in(0,1)$. This