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.
Categories: math.FA