Topological Noetherian operators: Generalized invertibility and additive representation
Let \(X\) and \(Y\) be Banach spaces. A bounded linear operator \(T: X\to Y\) is called Topological Noetherian if it has closed range and if both its kernel and range have closed complementary subspaces. It is shown that the operator \(T\) has a generalized inverse with respect to a pair of projection operators \(P: X\to X\) and \(Q: Y\to Y\) if and only if \(T\) is a Topological Noetherian operator. In that case \(\ker P\) and \(\ker Q\) will be complementary subspaces for \(\ker T\) and \(\text{ran } T\) respectively. The operator \(T\) is called Topological Fredholm if it is Topological Noetherian and if its range has a complementary subspace which is isomorphic to its kernel. A formula is derived for the generalized inverse with respect to \(P\) and \(Q\) for such an operator \(T\) in terms of any given isomorphism of \(\ker T\) onto the complement of its range \(\ker Q\).
scientific article; zbMATH DE number 794591