The aim of the present paper is to give some spectral results about generalized Fredholm operators and the so called S-generalized Fredholm operators, where S is a given bounded linear operator acting on a Banach space X. When X possesses some properties, we provide then some sufficient conditions for which a bounded linear operator will be a generalized Fredholm. The obtained results are applied to characterize the so called generalized S-essential spectrum, in particular the generalized Jeribi S-essential spectrum [17]. These results are formulated by means of measure of weak noncompactness.