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On the pseudospectrum of certain classes of non-self-adjoint pseudodifferential operators. (Sur le pseudo-spectre de certaines classes d’opérateurs pseudo-différentiels non auto-adjoints.) (French) Zbl 1201.35188

In this largely expository paper, the author studies pseudospectral properties for two classes of non self-adjoint pseudodifferential operators. We recall here that the \( \varepsilon \)-pseudospectrum for an operator in a given Hilbert space \(H\) is defined by \( \sigma _{\varepsilon }(A)=\{ z \in \mathcal{C} , \|(zI-A) ^{-1}\| \geq 1/ \varepsilon \}\) and that by a theorem of S.Roch and B.Silberman it is also equal to \(\cup \sigma (A+B)\) where the supremum is over all bounded operators \(B:H \rightarrow H\) of norm smaller than \( \varepsilon \). Pseudospectra, and this is visible from the second characterization, can for example influence the outcome of numerical calculations, in which \(A\) may be known only up to a perturbation, and can lead (by the first characterization) to large norms for the resolvents \((zI-A) ^{-1}\).
A significant part of the exposé deals with the author’s own paper [Duke Math. J. 145, No. 2, 249–279 (2008; Zbl 1157.35129)], and refers to the spectral stability of elliptic quadratic differential operators in terms of their Weyl symbol. In the last part of the paper, the author describes/studies spectral (in)stability for one-dimensional pseudodifferential operators which can be locally approximated by quadratic non-normal elliptic operators (which is given by the Weyl quantization of the Hessian of the principal symbol at a critical point). The results have appeared in the paper [Int. Math. Res. Not. 2007, No. 9, Article ID rnm029, 31 p. (2007; Zbl 1135.47048)].

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35P05 General topics in linear spectral theory for PDEs
47G30 Pseudodifferential operators
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