Embedding of Toeplitz operators with smooth symbols into strongly continuous semigroups

Using the model theory for Toeplitz operators with smooth symbols developed by the fourth author in the 80's, we study whether such operators $T_{F}$ can be embedded into a $C_{0}$-semigroup of operators on the Hardy space $H^p$ of the open unit disk, $1<p<\infty$. We show that it is the case as soon as $0$ belongs to the unbounded connected component of $\mathbb{C}$ minus the interior of the spectrum of $T_{F}$. We provide several conditions on the symbol $F$, both geometric and analytic in nature, ensuring that this sufficient condition is also necessary. For a certain class of symbols, where the curve $F(\mathbb{T})$ is a ``figure eight in a loop" such that $\mathbb{C}\setminus\sigma(T_F)$ has a bounded connected component, we obtain a complete characterization of the embeddability of $T_F$ into a $C_0$-semigroup. In the last part of the paper, we discuss the embeddability of $T_F$ when the symbol $F$ is not necessarily smooth, using connections with the numerical range and the functional calculus for bounded sectorial operators.