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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 THEOREM3

are norm bounded, every compact operator is AM-compact. An operator T is said to

dominate an operator S if |Sx| ? T|x| holds for all x ∈ E.

Theorem 3 ([3]). If C is a collection of positive operators on a Banach lattice X such

that

(i) C is finitely quasinilpotent at some positive non-zero vector, and

(ii) some operator in C dominates a non-zero AM-compact operator,

then C and [C? have a closed invariant order ideal.

Observe that if a collection C of positive operators has a (closed nontrivial) invariant

ideal then there exist non-zero positive x and f such that ?f,Tx? = 0 for all T ∈ C.

The converse is also true when C is a semigroup.

The goal of this paper is to “quantize” Theorem 3 in the same manner that Theo-

rem 1 was “quantized” into Theorem 2. Our proofs use ideas from [6] and [9].

In the following lemma, we collect several standard facts that we will use later. See,

e.g., [1] for the proofs.

Lemma 4. Let Z be a vector lattice, x ∈ Z+. Then for each y,z ∈ Z one has

(i) |x ∧ y − x ∧ z| ? |y − z|;

(ii) if |y| ? z then |x − x ∧ z| ? |x − x ∧ y|;

(iii) |x − x ∧ y| ? |x − y|.

From now on, X will be a real Banach lattice. We will also assume that X is a dual

Banach space; that is, X = Y∗for some (fixed) Banach space Y . We will start with a

version of Theorem 2 for convex collections of positive operators.

Theorem 5. Let C be a convex collection of positive adjoint operators on X. If there

is x0> 0 such that every operator from C is locally quasinilpotent at x0then there exist

non-zero x ∈ X+and f ∈ X∗

+such that ?f,Tx? ? ?T∗?efor all T ∈ C.

Remark 6. One might try to deduce Theorem 5 from Theorem 2 by considering the

W∗OT-closed algebra generated by C. However, the example in [5] shows that there

exists an algebra of nilpotent operators on a Hilbert space H which is WOT-dense in

L(H).

Proof of Theorem 5. Clearly, we may assume that ?x0? = 1. Also, without loss of

generality, C is closed under taking positive multiples of its elements, otherwise we

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

replace C with?αT : T ∈ C,0 < α ∈ R?. Fix 0 < ε <

1

10. Define

Cε =

?T ∈ C: ?T∗?e< ε?and

?z ∈ X: |z| ? Tx for some T ∈ Cε

Hε(x) =

?,x ∈ X+.

Then Hε(x) is convex and solid for all x ∈ X+.

Suppose that Hε(x) ?= X for some nonzero x ∈ X+. Since Hε(x) is convex, there is

a nonzero g ∈ X∗such that g(y) ? 1 for all y ∈ Hε(x). Consider h = |g| ∈ X∗. Then

for any y ∈ Hε(x) we have

h(y) ? h?|y|?= sup?g(u): − |y| ? u ? |y|?? 1

since Hε(x) is solid. In particular, ?h,Tx? ? 1 for all T ∈ Cε.

Put f =ε

i.e., ?T∗?e = 0, then αT ∈ Cε for all 0 < α ∈ R. Therefore ?h,αTx? ? 1 for all

0 < α ∈ R, so that ?f,Tx? = ?h,Tx? = 0. If T is not compact then

whence

2h. We claim that ?f,Tx? ? ?T∗?efor each T ∈ C. Indeed, if T is compact,

εT

2?T∗?e∈ Cε,

?f,Tx? = ?T∗?e

?

h,

εT

2?T∗?ex

?

? ?T∗?e.

Suppose now that Hε(x) = X for all nonzero x ∈ X+. Then, in particular, for each

x ∈ X there is yx∈ Hε(x) such that ?x0−yx? < ε. Fix an operator Tx∈ Cεsuch that

|yx| ? Txx. Then (ii) and (iii) of Lemma 4 yield ?x0− x0∧ Txx? < ε.

Let U0=?x ∈ X+: ?x−x0? ?1

operator Kx ∈ K(X) such that ?Kx− Tx? < ε. As compact adjoint operators are

w∗-?·? continuous on norm bounded sets, it follows that there is a relative (to U0)

w∗-open neigborhood Wx⊆ U0of x such that ?Kxz − Kxx? < ε whenever z ∈ Wx.

Then for every y ∈ Wxwe have:

2

?. Since

??(Tx)∗

??

e< ε, there is an adjoint compact

??x0− x0∧ Txy?????x0− x0∧ Txx??+??x0∧ Txx − x0∧ Kxx??

+??x0∧ Kxx − x0∧ Kxy??+??x0∧ Kxy − x0∧ Txy??

???x0− x0∧ Txx??+ ?Txx − Kxx? + ?Kxx − Kxy? + ?Kxy − Txy?

< ε + ε?x? + ε + ε?y? < 5ε <1

2.

Together with Tx? 0 this yields (x0∧ Txy) ∈ U0for each y ∈ Wx.

Note that U0is w∗-compact since U0is the intersection of X+with a closed ball.

Hence, we can find x1,...,xn∈ U0such that U0=?n

Txn∈ C. Then by Lemma 4(ii), we have x0∧ Tx ∈ U0for every x ∈ U0.

k=1Wxk. Define T = Tx1+ ··· +

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Define a sequence (yn) ⊆ U0by y0= x0and yn+1= x0∧ Tyn. Clearly 0 ? ynfor

all n, and yn? Tyn−1? ... ? Tny0, so that ?yn? ? ?Tnx0?. Thus yn→ 0 as n → ∞

by local quasinilpotence. This is a contradiction by the definition of U0.

?

The next theorem shows that the conclusion of Theorem 5 is also true for some

collections of operators which are not necessarily convex. We will, however, use a more

restrictive quasinilpotence condition. We will need some additional definitions.

Let C be a collection of positive operators. Following [1], define

DC=

?

D ∈ L(X)+ : ∃T1,...,Tk∈ [C? and

S1,...,Sk∈

∞

?

n=1

Cnsuch that D ?

k

?

i=1

TiSi

?

In other words, DCis the smallest additive and multiplicative semigroup which contains

the collection [C? · C and such that T ∈ DCand 0 ? S ? T imply S ∈ DC(see [1]).

Let C be a collection of positive adjoint operators on X. Define

EC=?T ∈ DC: T = S∗for some S ∈ L(Y )?.

Since adjoint operators are stable under addition and multiplication, ECis an additive

and multiplicative semigroup. It is also clear that C ⊆ EC.

Theorem 7. Let C be a collection of positive adjoint operators on X. If C is finitely

quasinilpotent at some x0> 0 then there exist non-zero x ∈ X+and f ∈ X∗

?f,Tx? ? ?T∗?efor all T ∈ EC.

+such that

Proof. Clearly ECis convex. Note that the finite quasinilpotence of C at x0implies the

finite quasinilpotence of DC(and, therefore, of EC) at x0(see, e.g., [1, Lemma 10.4]).

Finally, apply Theorem 5 to EC.

?

Now suppose, in addition, that Y is itself a Banach lattice. Then we can improve

the conclusion of Theorem 5. For an operator T acting on Y , define

θ(T) = inf??T − K?: K is AM-compact?.

Clearly, θ is a seminorm on L(Y ) and θ(T) = 0 if and only if T is AM-compact (because

the subspace of AM-compact operators in L(Y) is norm closed).

For ξ ∈ Y+, define a seminorm ρξon X via ρξ(x) = |x|(ξ).

Lemma 8. If ξ ∈ Y+and K ∈ L(Y ) is AM-compact, then K∗: (BX,w∗) → (X,ρξ) is

continuous.