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Dichotomy and \(S^ \infty\) functional calculi. (English) Zbl 0827.47033
Summary: Dichotomy for the abstract Cauchy problem with any densely defined closed operator on a Banach space is studied. We give conditions under which an operator with an \(H^\infty\) functional calculus has dichotomy. For the operators with imaginary axis contained in the resolvent set and with polynomial growth of the resolvent along the axis we prove the existence of dichotomy on subspaces and superspaces. Applications to the dichotomy of operators on \(L_p\)-spaces are given. The principle of linearized instability for nonlinear equations is proved.
MSC:
47D06 One-parameter semigroups and linear evolution equations
47A60 Functional calculus for linear operators
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