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The pseudospectra (or spectral instability) of non-selfadjoint operators is atopic of current interest in applied mathematics. In fact, for non-selfadjointoperators the resolvent could be very large outside the spectrum, making thenumerical computation of the complex eigenvalues very hard. This hasimportance, for example, in quantum mechanics, random matrix theory and fluiddynamics.  The occurence of pseudospectra for non-selfadjoint semiclassical differentialoperators is due to the existence of quasimodes, i.e., approximate localsolutions to the eigenvalue problem. For scalar operators, the quasimodesappear since the bracket condition is not satisfied for topological reasons,see the paper by Dencker, Sjostrand and Zworski in Comm. Pure Appl. Math. 57(2004), 384-415.  In this paper we shall investigate how these result can be generalized tosquare systems of semiclassical differential operators of principal type. Theseare the systems whose principal symbol vanishes of first order on its kernel.We show that the resolvent blows up as in the scalar case, except in a nowheredense set of degenerate values. We also define quasi-symmetrizable systems andsystems of subelliptic type for which we prove estimates on the resolvent.

The Pseudospectrum of Systems of Semiclassical Operators