The invariant subspace problem

Abstract

A fundamental problem is to determine whether every bounded linear transformation in Hilbert space has a nontrivial invariant subspace. A formal proof [1] of the existence of invariant subspaces is given by the theory of square summable power series [2] in its vector formulation [3]. A determination of extreme points of a convex set remains for the justification of the formal argument. A characterization of extreme points which implies the existence of invariant subspaces has been conjectured [4]. New information is obtained from a localization of the theory of square summable power series [5] which allows the formulation of extreme point problems which are closely related because of the Carathéodory-Fejér extension theorem [6]. The conjectured characterization of extreme points is shown to be false. Extreme points need not have the properties required for the construction of invariant subspaces.