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

(Impact Factor: 0.68). 05/2008; 137(05):1793-1800. DOI: 10.1090/S0002-9939-08-09775-X

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.

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• ##### Article: A remark on invariant subspaces of positive operators
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ABSTRACT: If S, T, R, and K are non-zero positive operators on a Banach lattice such that S ↔ T ↔ R ≤ K, where "↔" stands for the commutation relation, T is non-scalar, and K is compact, then S has an invariant subspace.
Proceedings of the American Mathematical Society 12/2013; 141(12). DOI:10.1090/S0002-9939-2013-11709-0 · 0.68 Impact Factor