A remark on invariant subspaces of positive operators

Abstract

If S S , T T , R R , and K K are non-zero positive operators on a Banach lattice such that S ↔ T ↔ R ⩽ K S\leftrightarrow T\leftrightarrow R\leqslant K , where “ ↔ \leftrightarrow ” stands for the commutation relation, T T is non-scalar, and K K is compact, then S S has an invariant subspace.