Criterion of reducibility for operator equations (Q1901907)

An equation (1) \(\Phi_x = 0\), where the operator \(\Phi\) is defined on a Banach space \(W\), is called reducible if there exists a completely continuous operator \(P : W \to W\) such that the solution set of the equation \(x = Px\) coincides with the solution set of (1). The author proves that (1) is reducible in this sense if and only if its solution set is closed and each bounded subset is pre-compact.