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Fredholm-type backstepping transformation, introduced by Coron and L\"u, hasbecome a powerful tool for rapid stabilization with fast development over thelast decade. Its strength lies in its systematic approach, allowing to deducerapid stabilization from approximate controllability. But limitations with thecurrent approach exist for operators of the form $|D_x|^\alpha$ for $\alpha \in(1,3/2]$. We present here a new compactness/duality method which hinges onFredholm's alternative to overcome the $\alpha=3/2$ threshold. More precisely,the compactness/duality method allows to prove the existence of a Riesz basisfor the backstepping transformation for skew-adjoint operator verifying$\alpha>1$, a key step in the construction of the Fredholm backsteppingtransformation, where the usual methods only work for $\alpha>3/2$. Theillustration of this new method is shown on the rapid stabilization of thelinearized capillary-gravity water wave equation exhibiting an operator ofcritical order $\alpha=3/2$.

Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves