Some Results in Fredholm Theory via The Measure of Noncompactness

Abstract Let A be a bounded linear operator in a complex Banach space X. We show that Id X − A is a Fredholm operator provided that A has a sufficiently small polynomially measure of noncompactness. In our general framework, we note that the case of Riesz operator becomes a particular one as it is for the other results in the domain. This enable us to obtain a new characterization for the Weyl essential spectrum of a closed densely defined operators. Mathematics Subject Classification: 47A53, 47A5