A Theorem of Krein Revisited

M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T ∗ has an eigenvector. We present generalizations of this result as well as some applications to C∗-algebras, operators on l1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras. M. Krein proved in [KR48, Theorem 3.3] that if T is a continuous operator on a normed space leaving invariant a non-empty open cone, then its adjoint T ∗ has an eigenvector. Krein’s result has an immediate application to the Invariant Subspace Problem because of the following observation. If T is a bounded operator on a Banach space and not a multiple of the identity, and T f = λf , then the kernel of f is a closed non-trivial subspace of codimension 1 which is invariant under T . Moreover, Range(λI − T ) is a closed nontrivial subspace which is proper (it is contained in the kernel of f) and hyperinvariant for T , that is, it is invariant under every operator commuting with T . Several proofs and modifications of Krein’s theorem appear in the literature, see, e.g., [AAB92, Theorems 6.3 and 7.1] and [S99, p. 315]. We prove yet another version of Krein’s Theorem: if T is a positive operator on an ordered normed space in which the unit ball has a dominating point, then T ∗ has a positive eigenvector. We deduce the original Krein’s version of the theorem from this, as well as several applications and related results. In particular, we show that if a bounded operator T on a Banach space satisfies any of the following conditions, then T ∗ has an eigenvector. Moreover, if the condition holds for a commutative family of operators, then the family of the adjoint operators has a common eigenvector. • T leaves invariant a cone with an interior point; • T is a positive operator on a C-algebra; 1991 Mathematics Subject Classification. Primary: 46B40, 47B60, 47B65; Secondary: 47A15, 47B48, 46L05, 46L10.