Some New Properties in Fredholm Theory, Schechter Essential Spectrum, and Application to Transport Theory

The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish a fine description of the Schechter essential spectrum of closed densely defined operators. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.