An overview of some recent developments on the Invariant Subspace Problem

Abstract

This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read's Banach space counter-example involving a finitely strictly singular operator. History of the problemThere is an important problem in operator theory called the invariant subspace problem. This problem is open for more than half a century, and there are many significant contributions with a huge variety of techniques, making this challenging problem so interesting; however the solution seems to be nowhere in sight. In this paper we first review the history of the problem, and then present an account of some recent developments with which we have been involved. Other recent approaches are discussed in [21]. The invariant subspace problem is the following: let be a complex Banach space of dimension at least 2 and let T ∈ ( ), i.e., T : → , linear and bounded. Is there any closed subspace ⊂ such that T ( ) ⊂ (i.e., is invariant for T ) and = {0} = (i.e., is not trivial)? In the sequel a "nontrivial invariant subspace" may be abbreviated as "ntis".• Here is a list of preliminary remarks:Assume that is of finite dimension ≥ 2, so that is isomorphic to C . Then T ∈ ( ) is a × matrix with complex entries, and thus T has eigenvectors. Each eigenvector generates a (nontrivial) invariant subspace of dimension 1. The Jordan form of T is of great help in describing the lattice of the invariant subspaces of T . Ifis not separable, then T ∈ ( ) has a nontrivial invariant subspace since for all ∈ , = 0, the closure of { (T ) : ∈ C[ ]} is invariant for T and nontrivial since it is a separable subspace by construction, containing a nonzero vector.Note also that for real Banach spaces, there exist operators with no nontrivial invariant subspace: take = R 2 and T a rotation of angle θ ∈ (0 2π) \