A REMARK ON INVARIANT SUBSPACES OF POSITIVE OPERATORS

Abstract. If S, T, R, and K are non-zero positive operators on a Banach lattice such that S ↔ T ↔ R 6 K, where “↔ ” stands for the commutation relation, T is non-scalar, and K is compact, then S has an invariant subspace. Throughout this note, X is a (real or complex) Banach lattice. For two operators S and T on X, the notation S ↔ T means that S and T commute. A (norm closed) subspace Y of X is said to be invariant under an operator T in L(X) if {0} 6 = Y 6 = X and TY ⊆ Y. We follow the notations and terminology of [AA02]. There have been many extensions of Lomonosov’s theorem [Lom73] to positive operators; see Chapter 10 of [AA02] for a review of the subject. In particular, if T ↔ R> K for some positive non-zero operators T, R, and K with T quasinilpo-tent and K compact, then T has an invariant subspace (even an invariant closed ideal). The condition T ↔ R> K can be replaced with T ↔ R 6 K or, even more generally, with T ↔ R> C 6 K for some non-zero positive operator C; in the latter case, T is said to be compact friendly. There have been several more re-cent similar extensions of Lomonosov’s theorem to positive quasinilpotent operators