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arXiv:0807.3327v1 [math.FA] 21 Jul 2008

A VERSION OF LOMONOSOV’S THEOREM FOR COLLECTIONS

OF POSITIVE OPERATORS

ALEXEY I. POPOV AND VLADIMIR G. TROITSKY

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 ∈ X and f ∈ X∗such that ?f,Tx? = 0 for all T ∈ 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, one has

???f,Tx???? ?T∗?e for all T ∈ A. In this paper, we prove a similar extension (in

case of adjoint operators) of a result of R. Drnovˇ sek. 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 x ∈ X+and f ∈ X∗

for all T ∈ C.

+such that ?f,Tx? ? ?T∗?e

In this paper we use techniques which were recently developed for transitive algebras

to obtain analogous results for collections of positive operators on Banach lattices. Let

us first briefly describe these two branches of the Invariant Subspace research.

Transitive algebras. Suppose that X is a Banach space. A subspace Z of X is

said to be invariant under an operator T ∈ L(X) if {0} ?= Z ?= X and T(Z) ⊆

Z. The Invariant Subspace Problem deals with the question: “Which operators have

invariant subspaces?”. Lomonosov proved in [7] that an operator which commutes with

a compact operator has an invariant subspace. There is also an algebraic version of

the problem: which subalgebras of L(X) have no (common) invariant subspaces? Such

subalgebras are called transitive. The classical Burnside’s theorem asserts that if X

is finite-dimensional then L(X) has no proper transitive subalgebras (clearly, L(X)

itself is always transitive). Using Lomonosov’s technique, Burnside’s theorem can be

extended to the infinite-dimensional case as follows:

Theorem 1 ([11, Theorem 8.23]). A proper WOT-closed subalgebra of L(X) contain-

ing a compact operator is not transitive.

A “quantitative” version of the later theorem was obtained by Lomonosov in [8] for

algebras of adjoint operators. Before we state it, we need to introduce some notation.

It is easy to see that a subalgebra A of L(X) has an invariant subspace if and only if

there exist non-zero x ∈ X and f ∈ X∗such that ?f,Tx? = 0 for every T ∈ A. Now

Date: July 21, 2008. Draft.

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2 A. I. POPOV AND V. G. TROITSKY

suppose that X is a dual space; that is, X = Y∗for some Banach space Y . If T ∈ L(X)

is a bounded adjoint operator on X then there is a unique operator S ∈ L(Y ) such

that S∗= T. We will write S = T∗; there will be no ambiguity as T∗ will always

be taken with respect to Y . We will write ?T?efor the essential norm of T, i.e., the

distance from T to the space of compact operators. Note that in general, for an adjoint

operator T, one has ?T?e? ?T∗?e. See [2] for an example of T such that ?T?e< ?T∗?e.

Theorem 2 ([8]). Let X be a dual Banach space and A a proper W∗OT-closed sub-

algebra of L(X) consisting of adjoint operators. Then there exist non-zero x ∈ X and

f ∈ X∗such that

???f,Tx???? ?T∗?efor all T ∈ A.

Invariant ideals of collections of positive operators. Suppose now that X is

a Banach lattice. Recall that a linear (not necessarily closed) subspace J ⊆ X is

called an order ideal if it is solid, i.e., y ∈ J implies x ∈ J whenever |x| ?

|y|. The following version of Lomonosov’s theorem for positive operators was proved

by B. de Pagter [10]: a positive quasinilpotent compact operator on X has a closed

invariant order ideal. There have been many extensions of this result, see, e.g., [1].

In particular, R. Drnovˇ sek [3] showed that a collection of positive operators satisfying

certain assumptions has a (common) invariant closed ideal. To state his result precisely,

we need to introduce more notations.

As usual, we write X+, X∗

and L(X), respectively. Let C be a collection of positive operators on X. Follow-

+, and L(X)+for the cones of positive elements in X, X∗,

ing [1], we will denote by symbols ?C] and [C? the super left and the super right

commutants of C, respectively, i.e.,

?C] =?S ∈ L(X)+: ST ? TS for each T ∈ C?,

[C? =?S ∈ L(X)+: ST ? TS for each T ∈ C?.

If D is another collection of operators then we write CD = {TS: T ∈ C,S ∈ D}. The

symbol Cnis defined as the product of n copies of C.

An operator T is locally quasinilpotent at x if limsupn?Tnx?

subset of X then we write ?U? = sup??x?: x ∈ U?. We call a collection C of operators

finitely quasinilpotent at a vector x ∈ X if limsupn?Fnx?

subcollection F of C. Clearly, finite quasinilpotence at x implies local quasinilpotence

at x of every operator in the collection.

If E is a Banach lattice then an operator T : E → E is called AM-compact if

1

n = 0. If U is a

1

n = 0 for every finite

the image of every order interval under T is relatively compact. Since order intervals

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A VERSION OF LOMONOSOV’S THEOREM9

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E-mail address: apopov@math.ualberta.ca

E-mail address: vtroitsky@math.ualberta.ca