A symbolic calculus for analytic Carleman classes

Abstract:

Let ${\mathcal {C}_M}\left ( {{I_\alpha }} \right )$ be the analytic Carleman class of ${\mathcal {C}^\infty }$-functions $f$ defined in a sector ${I_\alpha } = \left \{ {z \in {\mathbf {C}}:|\arg z| \leqslant \alpha \pi /2} \right \} \cup \left \{ 0 \right \}\left ( {0 \leqslant \alpha \leqslant 1} \right )$ and analytic in its interior such that ${\left \| {{f^{\left ( n \right )}}} \right \|_\infty } \leqslant C{\lambda ^n}{M_n}\left ( {n \geqslant 0} \right ),C = C\left ( f \right ),\lambda = \lambda \left ( f \right )$. In this paper, we give necessary and sufficient conditions in order that ${\mathcal {C}_M}\left ( {{I_\alpha }} \right )$ be inverse-closed. As a corollary, we obtain a characterization of ${\mathcal {C}_M}\left ( {{{\mathbf {R}}_ + }} \right )$ as an inverse-closed algebra, thus establishing the converse of a theorem of Malliavin [4] for the half-line.