A version of Lomonosov’s theorem for collections of positive operators

Abstract

It is known that for every Banach space X X and every proper W O T WOT -closed subalgebra A \mathcal A of L ( X ) L(X) , if A \mathcal A contains a compact operator, then it is not transitive; that is, there exist non-zero x ∈ X x\in X and f ∈ X ∗ f\in X^* such that ⟨ f , T x ⟩ = 0 \langle f,Tx\rangle =0 for all T ∈ A T\in \mathcal A . In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this result as follows: without having a compact operator in the algebra, one has | ⟨ f , T x ⟩ | ≤ ‖ T ∗ ‖ e \bigl \lvert \langle f,Tx\rangle \bigr \rvert \le \lVert T_*\rVert _e for all T ∈ A T\in \mathcal A . In this paper, we prove a similar extension of a result of R. Drnovšek. Specifically, we prove that if C \mathcal C is a collection of positive adjoint operators on a Banach lattice X X satisfying certain conditions, then there exist non-zero x ∈ X + x\in X_+ and f ∈ X + ∗ f\in X^*_+ such that ⟨ f , T x ⟩ ≤ ‖ T ∗ ‖ e \langle f,Tx\rangle \le \lVert T_*\rVert _e for all T ∈ C T\in \mathcal C .