Demicompactness Properties for Uniformly Continuous Cosine Families

In this paper, we establish some properties for a uniformly continuous cosine family ( C ( t ) ) t ∈ R which has the property that C ( t ) is demicompact for some (resp. every) t > 0 . More precisely, we prove that this property is equivalent to the demicompactness of I - A where A is the infinitesimal generator of the uniformly continuous cosine family ( C ( t ) ) t ∈ R . The obtained result is used to study the spectral inclusion for a uniformly continuous cosine family for an upper semi-Fredholm spectrum. In addition, we give some perturbation results on demicompactness.