Bounded Symbols and One-Component Inner Functions

Let $\theta$ be a nonconstant inner function, we prove that every bounded truncated Toeplitz operator on the model space $K_{\theta }=H^2\ominus \theta H^2$ admits a bounded symbol if and only if $\theta$ is one-component, that is, if $\{z\in \mathbb{D}:\mathrm{\mid }\theta (z)\mathrm{\mid }<r\}$ is nonempty and connected for some $r\in (0,1)$. This establishes the previously open necessary direction and resolves a conjecture of Baranov, Chalendar, Fricain, Mashreghi, and Timotin. For every non-one-component inner function, we construct a sequence of norm-one rank-one truncated Toeplitz operators whose infimal $L^{\mathrm{\infty }}$-norms over all representing symbols tend to infinity. The quantitative lower bound is obtained by forming two inner divisors from angularly separated parts of the zero set and singular measure. The argument applies without restrictions on the Blaschke or singular parts of $\theta$. The order of this lower bound is sharp, already for a family of degree-two Blaschke products. A continuation argument for local inverse branches produces the required sequence of points. Consequently, every non-one-component model space supports a trace-class truncated Toeplitz operator without a bounded symbol. Combined with known equivalences, the characterization also yields geometric criteria for weak factorization and for continuous-symbol representations of compact truncated Hankel operators.