Spectral properties of weakly almost periodic cosine functions (Q1284726)

The spectral structure of the infinitesimal generator of a strongly continuous cosine function of linear bounded operators is investigated. For example, under the assumption that \(C\) is a strongly continuous uniformly bounded cosine function of linear bounded operators acting on a complex Banach space \(E\) and \(X: D(X)\subset E\to E\) denotes the infinitesimal generator of \(C\) the relations \[ p\sigma(X)\subset p\sigma(X')\quad\text{and}\quad r\sigma(X)\cup p\sigma(X)= p\sigma(X') \] (where \(p\sigma(X)\) and \(r\sigma(X)\) denote, respectively, point spectrum and residual spectrum of the operator \(X\)) are shown. Most theorems concern the almost periodic behaviour of applications generated, in various ways, by \(C\). Some ergodic properties of cosine functions are also discussed, mainly in four situations: in the general case of a Banach space \(E\), in the framework of a weakly sequentially complete Banach space \(E\), in the case of a dual of a Banach space, and in the case in which the range of the infinitesimal generator is closed. Moreover, the case when the Banach space \(E\) is reflexive is discussed. In the last section, a first approach to the analysis of connections between dynamical systems and cosine functions is presented.