Page 1

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.

1

Page 2

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

Page 3

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

Page 4

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+ ··· +

Page 5

A VERSION OF LOMONOSOV’S THEOREM5

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.

Page 6

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

Proof. Let xα

w∗

− → x, with xα, x ∈ BX. Write

ρξ

?K∗xα− K∗x?=??K∗xα− K∗x??(ξ) = sup

where A = K?[−ξ,ξ]?. By assumption, K is AM-compact, thus A is a ?·?-compact

set.

For ν ∈ A, fix αν such that

???xα− x,ν???<

that ?µ − ν? <ε

−ξ?ζ?ξ?xα− x,Kζ? = sup

ν∈A?xα− x,ν?,

ε

3whenever α ? αν. If µ ∈ Y is such

3then for α ? ανwe have

???xα− x,µ????ε

3?xα− x? +???xα− x,ν???<2ε

n ?

3+ε

3= ε.

Pick ν1,...,νn∈ A such that A ⊆

k=1B(νk,ε

3). Then for every α ? max?αν1,...,ανn

?

?

we must have ρξ(K∗xα− K∗x) < ε.

An operator T ∈ L(X) will be said w∗-locally quasinilpotent at a pair (x0,ξ0),

where x0∈ X and ξ0∈ Y , if

??Tnx0(ξ0)??

at x0then T is w∗-locally quasinilpotent at (x0,ξ0) for every ξ0∈ Y .

1

n→ 0. Clearly, if T is locally quasinilpotent

Theorem 9. Suppose that X = Y∗for some Banach lattice Y , and C is a convex

collection of positive adjoint operators on X. Suppose that there exists a pair (x0,ξ0) ∈

X+× Y+such that x0(ξ0) ?= 0 and every operator from C is w∗-locally quasinilpotent

at (x0,ξ0). Then there exist non-zero x ∈ X+and f ∈ X∗

for all T ∈ C.

+such that ?f,Tx? ? θ(T∗)

Proof. The proof of the theorem is similar to that of Theorem 5. We may assume that

?x0? = 1, ?ξ0? = 1, and C is closed under taking positive multiples. Put ρξ0(x) =

|x|(ξ0). Evidently ρξ0(x) ? ?x? for all x ∈ X. It is also clear that |x| ? |y| implies

ρξ0(x) ? ρξ0(y).

Fix 0 < ε <

8

. Define

x0(ξ0)

Cε =

?T ∈ C: θ(T∗) < ε?and

?z ∈ X: |z| ? Tx for some T ∈ Cε

Gε(x) =

?,x ∈ X+.

Suppose that Gε(x) is not dense in X for some x ∈ X+. Analogously to the proof of

Theorem 5, we find a positive functional h ∈ X∗

Considering separately the cases θ(T∗) = 0 and θ(T∗) ?= 0, we get the conclusion of the

theorem.

Thus, we may assume that Gε(x) = X for all x > 0. Define

+such that ?h,Tx? ? 1 for all T ∈ Cε.

U0=?x ∈ X+: ?x? ? 1 and ρξ0(x − x0) ?x0(ξ0)

2

?.

Page 7

A VERSION OF LOMONOSOV’S THEOREM7

Clearly, U0is w∗-compact.

Let x ∈ U0be arbitrary. Since Gε(x) = X, we can find Tx∈ Cεsuch that ρξ0(x0−

x0∧ Txx) ? ?x0− x0∧ Txx? < ε. Fix an operator Kx adjoint to an AM-compact

operator such that ?Tx− Kx? < ε. By Lemma 8, we can find a relative (to U0) w∗-

open neighborhood Vx⊆ U0of x such that ρξ0(Kxx − Kxz) < ε for all z ∈ Vx. Then

for an arbitrary z ∈ Vx, we have

ρξ0

?x0− x0∧ Txz?? ρξ0

+ ρξ0

?x0− x0∧ Txx?+ ρξ0

?x0∧ Kxx − x0∧ Kxz?+ ρξ0

< ε + ?Txx − Kxx? + ρξ0(Kxx − Kxz) + ?Kxz − Txz?

< ε + ?Tx− Kx? · ?x? + ε + ?Tx− Kx? · ?z? < 4ε <x0(ξ0)

?x0∧ Txx − x0∧ Kxx?

?x0∧ Kxz − x0∧ Txz?

2

.

Take x1,...,xmin U0such that

m ?

k=1Vxk= U0. Then T = Tx1+ ··· + Txk∈ C satisfies

for all z ∈ U0. Since ?x0∧ Tz? ? ?x0? = 1, we have

ρξ0

x0∧ Tz ∈ U0for all z ∈ U0.

Put z0= x0and zn+1= x0∧ Tzn. By the w∗-local quasinilpotence of T at (x0,ξ0)

we have ρξ0(zn) ? ρξ0(Tnx0) =

??Tnx0(ξ0)??→ 0 as n → ∞ which is impossible by the

definition of U0.

?x0− x0∧ Tz?

?

x0(ξ0)

2

?

The following result is derived from Theorem 9 in the same way that Theorem 5 was

deduced from Theorem 7.

Theorem 10. Suppose that X = Y∗for some Banach lattice Y , and C is 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∗

+such that ?f,Tx? ? θ(T∗) for all T ∈ EC.

As every operator on ℓp(1 ? p < ∞) is AM-compact, this theorem can be used as

an alternative proof of the following (certainly known) result.

Corollary 11. Every collection of positive operators on ℓp, 1 < p < ∞, which is

finitely quasinilpotent at a non-zero positive vector, has a non-trivial closed common

invariant ideal.

Of course, Corollary 11 follows easily from Theorem 3 when 1 ? p < ∞.

Corollary 12. Every collection of positive adjoint operators on ℓ∞which is finitely

quasinilpotent at a non-zero positive vector has a non-trivial closed common invariant

ideal.

Page 8

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

The following example shows that the assumptions in Theorems 7 and 10 in general

do not guarantee the existence of an invariant subspace.

Example 13. There is a collection C of operators which satisfies all the conditions of

Theorem 10 and has no common non-trivial invariant subspaces. Namely, in [4], the

authors constructed a multiplicative semigroup Sp of positive square-zero operators

acting on Lp[0,1], 1 ? p < ∞, having no common non-trivial invariant subspaces. It is

not difficult to show that Spis in fact finitely quasinilpotent at every positive vector.

Hence for 1 < p < ∞, C = Spsatisfies the conditions of Theorem 10.

Remark 14. Even though Theorem 1 is not a special case of Theorem 2, in the case of

an algebra of adjoint operators the former can be easily deduced from the latter, see [8,

Corollary 1]. Similarly, we will show that in case of adjoint operators, Theorem 3 can

be deduced from Theorem 10. Indeed, suppose that X = Y∗for some Banach lattice Y ,

and C is a collection of positive adjoint operators which is finitely quasinilpotent at

some x0 > 0 and some operator in it dominates a non-zero AM-compact positive1

adjoint operator K. We will show that there is a non-trivial closed ideal which is

invariant under C and under all adjoint operators in [C?.

Clearly, K ∈ EC. Let x and f be as in Theorem 10.

J1 =

?z ∈ X : |z| ? T1KT2x for some T1,T2∈ EC

?z ∈ X : T|z| = 0 for all T ∈ EC

?z ∈ X : |z| ? Tx for some T ∈ EC

?,

J2 =

?, and

?.J3 =

It is easy to see that J1, J2, and J3 are ideals in X, invariant under C and under

all adjoint operators in [C?. It is left to show that at least one of the three must be

non-trivial. Clearly, J2is closed and J2?= X. Suppose that J2= {0}. In particular,

x / ∈ J2. It follows that J3 ?= {0}. Suppose that J3is dense in X. It follows from

Theorem 10 that J1 ⊆ kerf; hence J1 is proper. Assume that J1 = {0}. Hence,

T1KT2x = 0 for all T1,T2∈ EC. Since J2= {0}, it follows that K vanishes on ECx and,

therefore, on J3. Since J3is dense in X it follows that K = 0; a contradiction.

Acknowledgment. We would like to thank Victor Lomonosov for helpful discus-

sions.

1Unlike in Theorem 3, we also require that K ? 0 here.

Page 9

A VERSION OF LOMONOSOV’S THEOREM9

References

[1] Y.A. Abramovich, C.D. Aliprantis, An Invatation to Operator Theory, Graduate studies in math-

ematics, v.50.

[2] S. Axler, N. Jewell, A. Shields, The essential norm of an operator and its adjoint, Trans. Amer.

Math. Soc. 261 (1980), no. 1, 159–167.

[3] R. Drnovˇ sek, Common Invariant Subspaces for Collections of Operators, Integral Eq. Oper. Th.

39 (2001), 253–266.

[4] R. Drnovˇ sek, D. Kokol-Bukovˇ sek, L. Livshits, G. Macdonald, M. Omladiˇ c, and H. Radjavi, An

irreducible semigroup of non-negative square-zero operators, Integral Eq. Oper. Th. 42 (2002),

no. 4, 449–460.

[5] D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, A nil algebra of

bounded operators on Hilbert space with semisimple norm closure, Integral Eq. Oper. Th. 9

(1986), no. 5, 739–743.

[6] M. Lindstr¨ om and G. Schl¨ uchtermann, Lomonosov’s techniques and Burnside’s theorem, Canad.

Math. Bull. vol 43 (1), 2000 pp. 87-89.

[7] V. Lomonosov, Invariant subspaces of the family of operators that commute with a completely

continuous operator, Funktsional. Anal. i Prilozen 7 (1973), no. 3, 55-56 (Russian).

[8] V. Lomonosov, An Extension of Burnside’s Theorem to Infinite-Dimensional Spaces, Israel J.

Math, 75 (1991), 329-339.

[9] A.J. Michaels, Hilden’s simple proof of Lomonosov’s invariant subspace theorem, Adv. in Math.

25 (1977), 56-58.

[10] B. de Pagter, Irreducible compact operators, Math Z. 192 (1986), 149-153.

[11] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973.

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

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