On the spectrum of cosine operator functions (Q910692)

Let C(t), \(t\in {\mathbb{R}}\), be a strongly continuous cosine function, defined in a Hilbert space H, with infinitesimal generator A. Let \(\rho\) (A) and \(\sigma\) (A) denote the resolvent set and the spectrum of A respectively. It is known that for certain classes of cosine functions, including the class of uniformly bounded cosine functions, the following equivalence is true: \[ \mu \in \rho (C(t))\quad \Leftrightarrow \quad \{\lambda^ 2| ch \lambda t=\mu \}\subseteq \rho (A)\quad and\quad \sup_{ch \lambda t=\mu}\| \lambda (\lambda^ 2-A)^{-1}\| <\infty. \] In this paper the author proves that this is actually true for every strongly continuous cosine function defined in a Hilbert space.