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

TL;DR

This paper extends Lomonosov's theorem to collections of positive adjoint operators on Banach lattices, establishing the existence of non-zero positive vectors and functionals related to the essential norm of the operators.

Contribution

It provides a new extension of Lomonosov's theorem for collections of positive adjoint operators, generalizing previous results to broader settings.

Findings

Existence of non-zero positive x and f satisfying the inequality for all T in the collection.

Extension of Lomonosov's theorem to collections of positive operators on Banach lattices.

Conditions under which the inequality involving the essential norm holds.

Abstract

It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In the case of algebras of adjoint operators on a dual Banach space, V.Lomonosov extended this as follows: without having a compact operator in the algebra, |f(Tx)| is less than or equal to the essential norm of the pre-adjoint operator T_* for all T in A. In this paper, we prove a similar extension (in case of adjoint operators) of a result of R.Drnovsek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero positive x in X and f in X* such that f(Tx) is less than or equal to the essential norm of T_* for all T in C.