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We now define the notion of Shapiro–Lopatinskii ellipticity (also known as SLellipticity) of boundary conditions for an operator in Boutet de Monvel’s calculus on a smooth manifold X with boundary Y . The results can be found, for instance, in the monograph  of Rempel and Schulze, and of course, also in the work  of Boutet de Monvel; see also the monograph of Grubb . Therefore, here we only sketch the proofs.
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