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