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On the spectrum of cosine functions. (English) Zbl 0921.34073

The author gives some characterizations of the resolvent set ρ(A) of the generator A of a strongly continuous cosine function C(t) with the aid of the equation

u '' (t)=Au(t)+f(t)·

One of these theorems shows that 1ρ(C(1)) if and only if, for every 1-periodic function fC([0,1],X) (X is a Banach space), the above equation has a unique 1-periodic mild solution of class C 1 . In case of a Hilbert space X, the above equation has a unique 1-periodic mild solution for any 1-periodic function fC([0,1],X), if and only if, the associated sine function S(t) has the property that S(1) is invertible.

34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
47A25Spectral sets
34G10Linear ODE in abstract spaces
34C25Periodic solutions of ODE