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

The author gives some characterizations of the resolvent set $\rho \left(A\right)$ of the generator $A$ of a strongly continuous cosine function $C\left(t\right)$ with the aid of the equation

${u}^{\text{'}\text{'}}\left(t\right)=Au\left(t\right)+f\left(t\right)·$

One of these theorems shows that $1\in \rho \left(C\left(1\right)\right)$ if and only if, for every 1-periodic function $f\in C\left(\left[0,1\right],X\right)$ ($X$ 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 $f\in C\left(\left[0,1\right],X\right)$, if and only if, the associated sine function $S\left(t\right)$ has the property that $S\left(1\right)$ is invertible.

##### MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 47A25 Spectral sets 34G10 Linear ODE in abstract spaces 34C25 Periodic solutions of ODE