McIntosh formula for the gap between regular operators

Abstract

We derive an equivalent definition for the gap between two complemented submodules of a Hilbert $C^*$-module which is same as the one for closed subspaces of a Banach space. This gives an alternative way of defining gap between two regular operators. We give an alternative proof of the latter result. We also derive the McIntosh formula for computing the gap between two regular operators between Hilbert $C^*$-modules which is analogous to that of unbounded operators between Hilbert spaces.