On Generalized Inverses and Operator Ranges

Aspects of the theory of operator ranges, factorization and range inclusion are brought to bear on some operator and approximation-theoretic problems for generalized inverses on infinite dimensional Banach and Hilbert spaces. Several criteria are given for an operator to have a bounded outer inverse with infinite rank. It is also shown using one of these criteria that the set of all bounded linear operators with a bounded outer inverse is open. The set of all bounded linear operators with a bounded inner inverse is dense in the space of all bounded linear operators. Comments on related topics in generalized inverse operator theory and some open problems are given.