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arXiv · 2601.04146

Embedding of Toeplitz operators with smooth symbols into strongly continuous semigroups

Emmanuel Fricain·Sophie Grivaux·Maëva Ostermann·Dmitry Yakubovich
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Abstract

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 $\mathbb{C}\setminusσ(T_F)$ 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.

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Emmanuel Fricain, Sophie Grivaux, Maëva Ostermann, Dmitry Yakubovich. 2026-01-07. Embedding of Toeplitz operators with smooth symbols into strongly continuous semigroups. https://arxiv.org/abs/2601.04146

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