Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

Center Manifold Theory for Functional Differential Equations of Mixed Type

“…The last identity in the statement of the lemma follows easily by using the definition of ∆(z) and computing We conclude this section by referring the reader to [2,22], where similar results are obtained in the framework of delay equations.…”

Additive and Multiplicative Perturbations of Exponentially Dichotomous Operators on General Banach Spaces

“…where x ∈ H and 0 = t ∈ R. By assumption, in (3.5) the convolution kernel H(·; −S 0 )∆ is continuous in the norm except for a jump discontinuity in t = 0 and satisfies As a result [2], −S is exponentially dichotomous.…”

“…Such operators were introduced in [2,3] in connection with convolution equations on the half-line. Operators of this type also occur in various other applications, see, e.g., [7,8,9,10,11].…”

“…Perturbation results for exponentially dichotomous operators were already studied in [2], where additive perturbations were considered. Results in this direction were later obtained for more particular operators on Hilbert spaces in [8,9,10,11].…”

“…A closed and densely defined linear operator −S on a Banach space X is called exponentially dichotomous [2] if for some projection P commuting with S, the restrictions of S to Im P and of −S to Ker P are the infinitesimal generators of exponentially decaying C 0 -semigroups. We then define the bisemigroup generated by −S as…”