Article

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

08/2008;
Source: arXiv

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.

0 0
·
0 Bookmarks
·
73 Views
• Source
##### Article: The Essential Norm of an Operator and its Adjoint
Transactions of The American Mathematical Society - TRANS AMER MATH SOC. 01/1980; 261(1).
• Source
##### Article: Common invariant subspaces for collections of operators
[hide abstract]
ABSTRACT: Let C\mathcal{C} be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that C\mathcal{C} is finitely quasinilpotent at a vectorx 0F\mathcal{F} of C\mathcal{C} the joint spectral radius of F\mathcal{F} atx 0 is equal 0. If such collection C\mathcal{C} contains a non-zero compact operator, then C\mathcal{C} and its commutant C¢\mathcal{C}' have a common non-trivial invariant, subspace. If in addition, C\mathcal{C} is a collection of positive operators on a Banach lattice, then C\mathcal{C} has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let S\mathcal{S} be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then S\mathcal{S} has a common non-trivial invariant closed ideal.
Integral Equations and Operator Theory 08/2001; 39(3):253-266. · 0.71 Impact Factor
• ##### Article: An irreducible semigroup of non-negative square-zero operators
[hide abstract]
ABSTRACT: We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onL p [0,1), for 1p.
Integral Equations and Operator Theory 11/2002; 42(4):449-460. · 0.71 Impact Factor

1 Download
Available from