### Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems

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**Abstract :** We consider elliptic operators *A* on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of *A* through an observation, with an exponential cost. Following [LR95], we deduce the construction of a control for the non-selfadjoint parabolic problem *∂*_{t}u + Au = Bg. In particular, the *L*^{2} norm of the control that achieves the extinction of the lower modes of *A* is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of *A*.

**Mots Clés:** *Non-selfadjoint elliptic operators; Spectral theory; Spectral inequality; Coupled parabolic systems; Controllability.*

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