The perturbation of relatively open operators with reduced index

Abstract

AbstractLet X and Y denote normed spaces and T:D(T) ⊂ X → Y a linear transformation. It is shown that even in the case where both X and Y are incomplete, the quantity remains constant under both small and compact perturbation, provided that T is relatively open, R(T) is closed, and the perturbation is made in the right ‘direction’. If in addition and N(T) is topologically complemented, the topological complementation of the kernel is also preserved under small perturbations made in the right ‘direction’ and arbitrary compact perturbation. Various counter-examples are exhibited proving these results to be best possible.