On the collective compactness of strongly continuous semigroups and cosine functions of operators (Q1282139)

Let \(T(t)\), \(C(t)\), and \(S(t)\), \(t\in\mathbb{R}\) be the strongly continuous semigroup, the cosine function, and sine function of linear operators defined on a Banach complex space \(X\), and \(A\) be their infinitesimal generator. The author obtains fine criteria for the collective compactness of \(T(t)- I\), \(C(t)- I\), \(S(t)- I\) in terms of spectral properties. The proofs are based on several recently obtained criteria for almost periodicity, the asymptotic almost periodicity of \(T(t)\), \(C(t)\) and \(S(t)\). For example, the author establishes that the function \(T(t)-I\) is collectively compact (on every subset of \(t\in\mathbb{R}\)) if and only if the following conditions hold: (1) \(T(t)\) is uniform bounded and \(A\) is compact; (2) The spectrum \(\sigma(A)\) of \(A\) is a finite set of eigenvalues included in \(\{\lambda\in \mathbb{C}: \text{Re}(\lambda)\leq 0\}\); (3) The generalized eigenvectors of \(A\) span the space \(X\).