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We discuss (following mostly M. Putinar and J. Eschmeier) some applications of sheaves in the spectral theory of linear operators on Banach spaces.
In particular, we discuss the dictionary "operator with Bishop's property (\beta) = quasicoherent analytic Fr\'echet sheaf", "Foias decomposable operator = soft analytic Fr\'echet sheaf", "spectral subspace of an operator = section space of a sheaf"...