Upper-Lower and Left-Right Semi-Fredholmness

Abstract

Upper-lower and left-right approaches coincide for semi-Fredholm operators on Hilbert spaces but, in general, they are distinct for operators on a Banach space. The purpose of this paper is to investigate the difference between the upper-lower and left-right approaches for semi-Fredholm operators on a Banach space and, in particular, to verify when these approaches coincide. The program is based on the classes $\Gamma_R[\X]$ and $\Gamma_N[\X]$ of all operators on a Banach space $\X$ with complemented range and complemented kernel. It is shown that the intersection ${\Gamma_R[\X]\cap\Gamma_N[\X]}$ is algebraically and topologically large, and also that if $\Gamma_R[\X]$ and $\Gamma_N[\X]$ are either open, or closed, or if they coincide, then there is no difference between the upper-lower and left-right approaches for semi-Fredholm operators on a Banach space.