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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 ∂tu + Au = Bg. In particular, the L2 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.