On the quasi-Fredholm and Saphar spectrum of strongly continuous Cosine operator function

Let (C(t))_t∈𝕉 be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – coshλt is Saphar (resp. quasi-Fredholm) operator and λt /∉iπ𝕑, then A – λ² is also Saphar (resp. quasi-Fredholm) operator. We show by counter-example that the converse is false in general.