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abstract: |
    We consider a non-self-adjoint $h$-pseudodifferential operator $P$ in the semi-classical limit ($h\to 0$). If $p$ is the leading symbol, then under suitable assumptions about the behaviour of $p$ at infinity, we know that the resolvent $(z-P)^{-1}$ is uniformly bounded for $z$ in any compact set not intersecting the closure of the range of $p$. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance ${\cal O}(1)((h\ln \frac{1}{h})^{k/(k+1)})$ from certain boundary points, where $k\in \{ 2,4,...\}$. This is a slight improvement of a result by Dencker, Zworski and the author, and it has recently been obtained by W. Bordeaux Montrieux in a model situation where $k=2$. The method of proof is different from the one of Dencker et al, and is based on estimates of an associated semi-group.

    **Résumé**

    Nous consid' erons un op' erateur $h$-pseudodiff' erentiel non-autoadjoint $P$ dans la limite semi-classique ($h\to 0$). Si $p$ d' esigne le symbole principal, alors sous des hypothèses convenables sur le comportment de $p$ à l’infini nous savons que la r' esolvante $(z-P)^{-1}$ est uniform' ement born' ee pour $z$ dans un compact qui ne rencontre pas l’adh' erence de l’image de 