Maximal dominated operator semigroups

Semigroup Forum Vol. 64 (2002) 376–390c© 2002 Springer-Verlag New York Inc.

DOI: 10.1007/s002330010123

RESEARCH ARTICLE

Maximal Dominated Operator Semigroups

Roman Drnovsek and Matjaz Omladic

Communicated by Jimmie D. Lawson

Abstract

Maximal operator semigroups, bounded in a certain sense, on real or complexvector spaces are studied. For any maximal semigroup M dominated by acertain pair of homogeneous functions there is an operator quasinorm for whichM is exactly the semigroup of contractions in this quasinorm. Applications toRiesz spaces are given. In particular, maximal semigroups of matrices dominatedby a given positive matrix are characterized. We thus answer the questionimplicitly posed in [2].

Key words and phrases: operators, vector spaces, semigroups, contractions,domination, Riesz spaces.

2002 Mathematics Subject Classification: 47D03, 47B65.

1. Introduction

Let X be a vector space over the field of either real or complex numbers.By operator we mean a linear transformation from the vector space X intoitself. The vector space of all operators on X is denoted by L(X). Thisis also a semigroup for composition of operators as multiplication. In thispaper we are concerned with subsemigroups of L(X) that are bounded in acertain sense and maximal with respect to this bound. Maximal boundedsemigroups on finite-dimensional spaces X have been studied extensively, cf.[2] and the references given there. The semigroups dominated by a matrix ofcertain type, called semigroup patterns, were studied there. The study is closelyrelated to and motivated by the study of algebras with commutative lattice ofinvariant subspaces. The immediate generalization of these results to infinitedimensions would go to Riesz spaces, where one has a naturally extended notionof coordinatewise domination of an operator. However the results presented hereare substantially more general and hold in arbitrary vector spaces. Almost allof our results seem to be new even for the case of finite dimensions.

The main results and proofs are presented in Section 2. We introduce thenotion of domination of an operator by a pair of functions. It is not hard to seethat under some quite general technical conditions a semigroup of contractionsin a given operator quasinorm is dominated by a pair of homogeneous functions.It is somewhat more surprising that the converse is also true in some sense(Theorem 2.3), namely, that for a maximal semigroup M dominated by a

Drnovsek and Omladic 377

certain pair of homogeneous functions there exists an operator quasinorm forwhich M is exactly the semigroup of contractions in this quasinorm. Wealso give technical conditions under which the obtained operator quasinormis induced by norms (Theorem 2.8).

Section 3 is devoted to applications of these results to Riesz spaces. Wegive a useful characterization of the absolute domination of an order boundedoperator by a positive operator (Proposition 3.1) and a characterization ofa maximal semigroup of order bounded operators absolutely dominated by apositive operator (Theorem 3.2). The main result of this section (Theorem 3.4)represents a maximal semigroup of order bounded operators dominated by apositive operator of rank one as a semigroup of contractions in a certain operatorquasinorm. It also gives technical conditions under which this quasinorm isinduced by norms. A finite-dimensional case of our results gives an answer tothe question implicitly posed in [2] about the description of maximal semigroupsof matrices dominated by a given positive matrix.

2. The case of the general vector space

The vector space of all linear functionals on the vector space X , the dual spaceof X , is denoted by X ′ . The adjoint of an operator T ∈ L(X) is denoted byT ′ . A collection C of linear transformations from a vector space X to a vectorspace Y is said to separate points of X if for each non-zero vector x ∈ X thereexists an A ∈ C with Ax 6= 0. Given y ∈ X and ψ ∈ X ′ , we denote by y ⊗ ψthe operator on X defined by

(y ⊗ ψ)(x) = ψ(x) y , x ∈ X.

A function n: X → [0,∞] is said to be homogeneous if n(0) = 0 andn(λx) = |λ|n(x) for all scalars λ and for all x ∈ X with n(x) < ∞ . Ahomogeneous function ‖ · ‖: X → [0,∞] is called a quasinorm on X whenever‖x + y‖ ≤ ‖x‖ + ‖y‖ for all x, y ∈ X . If ‖x‖ < ∞ for every x ∈ X , then thequasinorm ‖ · ‖ is said to be a seminorm on X . If, in addition, ‖x‖ = 0 impliesthat x = 0, the seminorm is called a norm.

Let ‖ · ‖1 be a quasinorm on a vector space X , and let ‖ · ‖2 be aseminorm on X such that ‖x‖2 ≤ ‖x‖1 for all x ∈ X . The operator quasinorm||| · ||| induced by ‖ · ‖1 and ‖ · ‖2 is defined on the vector space L(X) by

|||T ||| := sup{‖Tx‖1: x ∈ X, ‖x‖2 ≤ 1}, T ∈ L(X).

It is clear that ‖Tx‖1 = 0 for any T ∈ L(X) with |||T ||| < ∞ and any x ∈ Xwith ‖x‖2 = 0. Consequently, we have the inequality ‖Tx‖1 ≤ |||T ||| · ‖x‖2 forall x ∈ X provided |||T ||| <∞ .

Proposition 2.1. Let S and T be operators on X with |||S||| <∞ . Then

|||ST ||| ≤ |||S||| · |||T |||.

378 Drnovsek and Omladic

Proof. If |||S||| = 0, then ‖Sx‖1 = 0 for all x ∈ X , so that ‖STx‖1 = 0 forall x . Therefore |||ST ||| = 0, and the desired inequality holds. If |||T ||| = 0, then‖Tx‖1 = 0 for all x ∈ X , and so ‖Tx‖2 = 0 for all x ∈ X . Since |||S||| <∞ , wenow conclude that ‖STx‖1 = ‖S(Tx)‖1 = 0 for all x ∈ X . Hence |||ST ||| = 0as claimed. If 0 < |||S||| and |||T ||| =∞ , then the inequality is trivially true.

We can therefore assume that 0 < |||S||| < ∞ and 0 < |||T ||| < ∞ . Thereis no loss in assuming that |||S||| = |||T ||| = 1. Since ‖Tx‖1 ≤ 1 for all ‖x‖2 ≤ 1,we have

|||ST ||| = sup{‖STx‖1: x ∈ X, ‖x‖2 ≤ 1} ≤ sup{‖S(Tx)‖1: x ∈ X, ‖Tx‖1 ≤ 1}.

It follows that

|||ST ||| ≤ sup{‖Sy‖1: y ∈ X, ‖y‖1 ≤ 1} ≤ sup{‖Sy‖1: y ∈ X, ‖y‖2 ≤ 1} = 1,

and the proof is finished.

An operator T ∈ L(X) is said to be a contraction in the operatorquasinorm ||| · ||| if |||T ||| ≤ 1. It follows from Proposition 2.1 that the set ofall contractions in the operator quasinorm ||| · ||| is a (multiplicative) semigroup.

Because of applications of our results to Riesz spaces we choose and fixa non-zero linear subspace Φ of the dual space X ′ . Furthermore, we also fix asubalgebra LΦ(X) of L(X) with the properties:

(a) the identity operator I on X belongs to LΦ(X);

(b) LΦ(X) contains all operators of the form x⊗ φ with x ∈ X and φ ∈ Φ;

(c) for any T ∈ LΦ(X) the adjoint T ′ leaves the subspace Φ invariant.

Definition 2.2. Given functions n: X → [0,∞) and n′: Φ → [0,∞), anoperator T on X is said to be dominated by the pair (n, n′) whenever

|φ(Tx)| ≤ n′(φ)n(x)

for all x ∈ X and for all φ ∈ Φ. A collection C of operators on X is said tobe dominated by the pair (n, n′) whenever every member of C is dominated by(n, n′).

Note first that there is no loss of generality in assuming that n and n′ arehomogeneous. If this is not true, define homogeneous functions n: X → [0,∞)and n′: Φ→ [0,∞) by

n(x) := infλ6=0

n(λx)

|λ| and n′(φ) := infλ6=0

n′(λφ)

|λ| .

It can be easily verified that an operator T on X is dominated by the pair(n, n′) whenever it is dominated by (n, n′).


Let ||| · ||| be an operator quasinorm on LΦ(X) induced by a quasinorm‖ · ‖1 and a seminorm ‖ · ‖2 . Assume further that there exists a seminorm ‖ · ‖′1on Φ such that ‖x‖1 = sup{|φ(x)| : φ ∈ Φ, ‖φ‖′1 ≤ 1} for all x ∈ X . Then thefunctions n: X → [0,∞) and n′: Φ→ [0,∞) defined by

n(x) = ‖x‖2 and n′(φ) = ‖φ‖′1

are homogeneous. Denote by C the semigroup of all contractions in ||| · ||| . Theneach T ∈ C is dominated by the pair (n, n′). Indeed, for all x ∈ X and for allφ ∈ Φ,

|φ(Tx)| ≤ ‖φ‖′1 ‖Tx‖1 ≤ ‖φ‖′1 |||T ||| ‖x‖2 ≤ ‖φ‖′1 ‖x‖2 = n′(φ)n(x).

On the other hand, if T ∈ LΦ(X) is dominated by the pair (n, n′), then, forany x ∈ X ,

‖Tx‖1 = sup{|φ(Tx)| : φ ∈ Φ, ‖φ‖′1 ≤ 1} ≤ n(x) = ‖x‖2,

so that |||T ||| ≤ 1, that is, T ∈ C . We thus conclude that C is a maximalsubsemigroup of LΦ(X) among those dominated by (n, n′).

The following main result of this paper shows that the converse of theabove observation holds.

Theorem 2.3. Assume that the functions n: X → [0,∞) and n′: Φ →[0,∞) are homogeneous. Let M be a maximal subsemigroup of LΦ(X) amongthose dominated by (n, n′) . Then there exist a seminorm ‖ · ‖′1 on Φ and aseminorm ‖ · ‖2 on X such that

(i) ‖x‖2 ≤ ‖x‖1 for all x ∈ X , where

‖x‖1 := sup{|φ(x)| : φ ∈ Φ, ‖φ‖′1 ≤ 1}

is a quasinorm on X ;

(ii) M = {T ∈ LΦ(X) : |||T ||| ≤ 1} , where

|||T ||| := sup{‖Tx‖1 : ‖x‖2 ≤ 1}.

If some non-zero multiple of the identity operator I on X is dominated by(n, n′) , then we can achieve that ‖ · ‖1 is a seminorm which is equivalent to theseminorm ‖·‖2 . If I is dominated by (n, n′) , then we can even take ‖x‖1 = ‖x‖2for all x ∈ X .

Proof. Define the quasinorms ‖ · ‖′ and ‖ · ‖′1 on Φ by

‖φ‖′ := sup{|φ(x)| : x ∈ X,n(x) ≤ 1}


and

‖φ‖′1 := sup{‖S′φ‖′ : S ∈M} = sup{|φ(Sx)| : S ∈M, n(x) ≤ 1}.

Since ‖φ‖′1 ≤ n′(φ), we have 0 ≤ ‖φ‖1 < ∞ for all φ ∈ Φ, so that ‖ · ‖′1 isa seminorm on Φ. Furthermore, ‖S′φ‖′1 ≤ ‖φ‖′1 for all S ∈ M and φ ∈ Φ.Indeed, since M is a semigroup, we have

‖S′φ‖′1 = sup{‖T ′S′φ‖′ : T ∈M} = sup{‖(ST )′φ‖′ : T ∈M}≤ sup{‖T ′φ‖′ : T ∈M} = ‖φ‖′1.

For every x ∈ X define

‖x‖1 := sup{|φ(x)| : φ ∈ Φ, ‖φ‖′1 ≤ 1}

and

‖x‖2 := sup{‖Sx‖1 : S ∈M} = sup{|φ(Sx)| : S ∈M, φ ∈ Φ, ‖φ‖′1 ≤ 1}.

Obviously, ‖ · ‖1 and ‖ · ‖2 are quasinorms on X . Observe that |φ(x)| ≤‖φ‖′ · n(x) for all x ∈ X and for all φ ∈ Φ with ‖φ‖′ < ∞ . In particular, if‖φ‖′1 ≤ 1, S ∈M and x ∈ X , then

|φ(Sx)| = |(S′φ)(x)| ≤ ‖S′φ‖′ · n(x) ≤ n(x),

because ‖S′φ‖′ ≤ ‖φ‖′1 ≤ 1. It follows that ‖x‖2 ≤ n(x) for all x ∈ X , so that0 ≤ ‖x‖2 <∞ for all x ∈ X . This shows that ‖ · ‖2 is a seminorm on X . Since‖φ‖′1 ≤ 1 implies that ‖S′φ‖′1 ≤ 1 for each S ∈M , we have

‖x‖2 = sup{|(S′φ)(x)| : S ∈M, φ ∈ Φ, ‖φ‖′1 ≤ 1}≤ sup{|ψ(x)| : ψ ∈ Φ, ‖ψ‖′1 ≤ 1} = ‖x‖1,

which proves (i).

By Proposition 2.1 the set of contractions C := {T ∈ LΦ(X) : |||T ||| ≤ 1}is a semigroup. Because ‖x‖2 ≤ n(x) for all x ∈ X and ‖φ‖′1 ≤ n′(φ) for allφ ∈ Φ, we have

|φ(Tx)| ≤ ‖φ‖′1 · ‖Tx‖1 ≤ ‖φ‖′1 · |||T ||| · ‖x‖2 ≤ n′(φ) · n(x)

for all T ∈ C , x ∈ X , and φ ∈ Φ, so that C is dominated by (n, n′). SinceM ⊆ C by the definition of the seminorm ‖ · ‖2 , the maximality of M nowimplies that M = C . Thus (ii) has been proved.

Suppose now that for some λ 6= 0 the operator λI on X is dominated by(n, n′). Since we may assume that 0 < λ ≤ 1, we have λI ∈M by maximalityof M . Then, for any x ∈ X ,

‖x‖2 = sup{‖Sx‖1 : S ∈M} ≥ λ ‖x‖1,


which implies that ‖ · ‖1 is a seminorm which is equivalent with the seminorm‖ · ‖2 . If we can even take λ = 1 in the last consideration, we obtain togetherwith (i) that ‖x‖1 = ‖x‖2 for all x ∈ X . This completes the proof of thetheorem.

It is natural to ask under what assumptions on the functions n and n′

the seminorms ‖ · ‖′1 and ‖ · ‖2 are actually norms.

Definition 2.4. A homogeneous function n: X → [0,∞) is said to be aweak function on X if there exists a non-zero functional ψ ∈ Φ such that

|ψ(x)| ≤ n(x)

for all x ∈ X . Similarly, a homogeneous function n′: Φ → [0,∞) is called aweak function on Φ if there exists a non-zero vector y ∈ X such that

|φ(y)| ≤ n′(φ)

for all φ ∈ Φ.

Lemma 2.5. Assume that n: X → [0,∞) and n′: Φ → [0,∞) are weakfunctions and let ψ ∈ Φ and y ∈ X be as in Definition 2.4. Furthermore, letM be a maximal subsemigroup of LΦ(X) among those dominated by the pair(n, n′) . Then there exists λ > 0 such that the operator A := λ y⊗ψ is in M .

Proof. Choose λ > 0 satisfying the following conditions

λ ≤ 1 , λ n′(ψ) ≤ 1 , λ n(y) ≤ 1 , λ n(y)n′(ψ) ≤ 1.

We want to show that the semigroup S generated by M and the operator A isstill dominated by the pair (n, n′). Then S =M by maximality of M , and soA ∈M . It is enough to show that if operators S, T ∈ LΦ(X) are dominated bythe pair (n, n′), then the operators A , AT , SA , and SAT are also dominatedby the pair (n, n′). With this end in view, choose x ∈ X and φ ∈ Φ. Then wehave

|φ(Ax)| = λ |φ(y)| |ψ(x)| ≤ n′(φ)n(x).

Using the fact that the operators S and T are dominated by (n, n′), we obtainthat

|φ(ATx)| = λ |φ(y)| |ψ(Tx)| ≤ λn′(φ)n′(ψ)n(x) ≤ n′(φ)n(x),

|φ(SAx)| = λ |φ(Sy)| |ψ(x)| ≤ λn′(φ)n(y)n(x) ≤ n′(φ)n(x),

and

|φ(SATx)| = λ |φ(Sy)| |ψ(Tx)| ≤ λn′(φ)n(y)n′(ψ)n(x) ≤ n′(φ)n(x).

The proof of the lemma is complete.


A set U of a vector space Y is absorbing if for every y ∈ Y there existsλ > 0 such that λy ∈ U .

Definition 2.6. A homogeneous function n: X → [0,∞) is said to be astrong function on X if there exists an absorbing subset U ′ of Φ such that

|ψ(x)| ≤ n(x)

for all x ∈ X and for all ψ ∈ U ′ . Similarly, a homogeneous function n′: Φ →[0,∞) is called a strong function on Φ if there exists an absorbing subset U ofX such that

|φ(y)| ≤ n′(φ)

for all φ ∈ Φ and for all y ∈ U .

Lemma 2.7. Assume that n: X → [0,∞) and n′: Φ → [0,∞) are weakfunctions. Let M be a maximal subsemigroup of LΦ(X) among those dominatedby the pair (n, n′) .

(a) If n is a strong function and if Φ separates points of X , then M separatespoints of X .

(b) If n′ is a strong function, then the dual semigroup M′ := {S′ : S ∈ M}separates points of Φ .

Proof. If n is a strong function on X , then there exists an absorbing subsetU ′ of Φ such that |ψ(x)| ≤ n(x) for all x ∈ X and for all ψ ∈ U ′ . Assume nowthat there is a non-zero vector x ∈ X such that Sx = 0 for all S ∈M . BecauseΦ separates points of X , there is a functional ψ ∈ Φ satisfying ψ(x) 6= 0. SinceU ′ is absorbing, there is no loss in assuming that ψ ∈ U ′ . Since the functionn′ is a weak function on Φ, there exists a non-zero vector y ∈ X such that|φ(y)| ≤ n′(φ) for all φ ∈ Φ. By Lemma 2.5 there exists λ > 0 such that theoperator A := λ y ⊗ ψ is in M . However, Ax = λψ(x) y 6= 0, which is in acontradiction with the above. This proves (a).

The proof of (b) is similar. If n′ is a strong function on Φ, then thereexists an absorbing subset U of X such that |φ(y)| ≤ n′(φ) for all φ ∈ Φ andfor all y ∈ U . Suppose that there is a non-zero functional φ ∈ Φ such thatS′φ = 0 for all S ∈ M . Since U is absorbing, there is a vector y ∈ U suchthat φ(y) 6= 0. Since n is the weak function on X , there exists a non-zerofunctional ψ ∈ Φ such that |ψ(x)| ≤ n(x) for all x ∈ X . By Lemma 2.5 thereexists λ > 0 such that the operator A := λ y ⊗ ψ is in M . Now,

(A′φ)(x) = φ(Ax) = λφ(y)ψ(x)

for all x ∈ X and ψ is a non-zero functional, so that A′φ 6= 0, contradictingthe above and therefore completing the proof.


We are now able to show the following complements of Theorem 2.3.

Theorem 2.8. In the situation of Theorem 2.3 assume that n: X → [0,∞)and n′: Φ→ [0,∞) are weak functions.

(a) If n is a strong function and if Φ separates points of X , then the semi-norm ‖ · ‖2 can be chosen to be a norm. If, in addition, some non-zeromultiple of the identity operator I on X is dominated by (n, n′) , then‖ · ‖1 is also a norm which is equivalent with the norm ‖ · ‖2 .

(b) If n′ is a strong function, then the seminorm ‖ · ‖′1 can be chosen to be anorm.

Proof.

(a) Let us prove that the seminorm ‖ · ‖2 from the proof of Theorem 2.3 isa norm on X . If ‖x‖2 = 0, then φ(Sx) = 0 for all S ∈ M and for allφ ∈ Φ. Since Φ separates points of X , we conclude that Sx = 0 for allS ∈M . Now, M separates points of X by Lemma 2.7 (a), and so x = 0.The second assertion is clear.

(b) Similarly, by an application of Lemma 2.7 (b), one can show easily thatif n′ is a strong function, then the seminorm ‖ · ‖′1 from the proof ofTheorem 2.3 is a norm on Φ.

Theorem 2.9. For each i from a given index set I , let the functions ni:X → [0,∞) and n′i: Φ→ [0,∞) be homogeneous. Assume that M is a maximalsubsemigroup of LΦ(X) among those dominated by (ni, n

′i) for all i ∈ I . Then

for each i ∈ I there exists an operator quasinorm ||| · |||i on LΦ(X) such that

M = {T ∈ LΦ(X) : ‖T‖ ≤ 1},

where ‖T‖ := sup{|||T |||i : i ∈ I} .

Proof. By Zorn’s Lemma for each i ∈ I there exists a maximal subsemi-group Mi of LΦ(X) among those dominated by (ni, n

′i). By Theorem 2.3, Mi

is equal to the semigroup Ci of all contractions in some operator quasinorm ||| · |||ion LΦ(X). Since M ⊆ Ci for all i , we have M ⊆ C := ∩i∈ICi . It is easy tosee that C = {T ∈ LΦ(X) : ‖T‖ ≤ 1} . Furthermore, C is a subsemigroup ofLΦ(X) which is dominated by (ni, n

′i) for all i ∈ I . Now, the maximality of

M yields M = C , as desired.

3. The case of the Riesz space

Let E be a real Riesz space, i.e., a partially ordered real vector space in whichthe supremum and infimum of each pair of elements exist. The positive part,the negative part and the absolute value (or the modulus) of x ∈ E are defined


by x+ = sup{x, 0} , x− = sup{−x, 0} and |x| = sup{x,−x} , respectively. Thepositive cone of E is denoted by E+ , that is, E+ := {x ∈ E : x ≥ 0} . Anelement u ∈ E+ is called a strong order unit whenever for each x ∈ E thereexists a λ > 0 with |x| ≤ λu . The Riesz space E is called Archimedean,whenever x, y ∈ E+ and nx ≤ y for all n ∈ N imply that x = 0. If everynon-empty subset which is bounded from above has a supremum, then E issaid to Dedekind complete. Recall that every Dedekind complete Riesz space isalso Archimedean. For any x , y ∈ E with x ≤ y the set {z ∈ E : x ≤ z ≤ y}is called the order interval between x and y . An operator T on E is calledpositive if Tx ≥ 0 for all x ≥ 0, and order bounded whenever it maps orderintervals into order intervals. By Lb(E) we denote the partially ordered vectorspace of all order bounded operators on E . Recall that Lb(E) is a Dedekindcomplete Riesz space whenever E is Dedekind complete. The Riesz space of allorder bounded linear functionals on E is denoted by E˜, and it is called theorder dual of E . In the Riesz space theory complex Riesz spaces are studied aswell. Under a complex Riesz space we understand the complexification E + iEof the real Riesz space E . The absolute value of x ∈ E + iE is defined by

|x| = sup(Re (x eiθ) : θ ∈ [0, 2π))

if this supremum exists. By [6, Theorem 91.2] this is true if E is Archimedeanand uniformly complete. In particular, the last assumption on E is satisfiedwhenever E is Dedekind complete.

Throughout the paper, let E a real or complex Riesz space with non-trivial order dual E˜. In the case of a complex Riesz space we assume that forany x ∈ E the supremum in the definition of the absolute value |x| exists. Fordetails about Riesz spaces not explained above, we refer the reader to the books[3], [6], [5], [1], and [4].

Let T , K be order bounded operators on a Riesz space E with Kpositive. We say that T is absolutely dominated by K if the modulus |T |exists and |T | ≤ K . We also say that T is dominated by K whenever

|φ(Tx)| ≤ |φ|(K|x|)

for all x ∈ E and φ ∈ E˜.Proposition 3.1. Let T and K be order bounded operators on E with Kpositive. Consider the following assertions:

(a) T is absolutely dominated by K ;

(b) |Tx| ≤ K|x| for all x ∈ E ;

(c) T is dominated by K ;

(d) |φ(Tx)| ≤ φ(K|x|) for all x ∈ E and φ ∈ (E )+ .


Then (a) ⇒ (b) ⇒ (c) ⇒ (d) . If E˜ separates points of E , then (d) implies(b). If E is Dedekind complete, then (b) implies (a). In particular, if E isDedekind complete and E˜ separates points of E , then all four assertions areequivalent.

Proof. If (a) holds, then |Tx| ≤ |T ||x| ≤ K|x| for all x ∈ E , and so (b) istrue. If we assume that (b) holds, then

|φ(Tx)| ≤ |φ|(|Tx|) ≤ |φ|(K|x|)

for all x ∈ E and φ ∈ E˜, so that (c) holds. The implication (c)⇒ (d) is clear.

Assume now that E˜ separates points of E . To show that (d) implies(b), let x ∈ E . Then, for any φ ∈ (E )+ ,

φ(K|x|) ≥ Re (φ(Tx) eiθ)

for any θ ∈ [0, 2π) in the complex case, and for θ ∈ {0, π} in the real case. Thisimplies that

φ(K|x| − Re (eiθ Tx)) ≥ 0

for each φ ∈ (E )+ . Since E˜ separates points of E , an application of [1,Theorem 5.1] (that clearly holds in the complex case as well) gives that

K|x| − Re (eiθ Tx) ≥ 0

for all θ , which yields

K|x| ≥ supθ

(Re (eiθ Tx)) = |Tx|.

Finally, let us show that (b) implies (a) provided E is Dedekind complete.In this case the modulus |T | exists in Lb(E). For each x ∈ E+ and for eachy ∈ E with |y| ≤ x we have |Ty| ≤ K|y| ≤ Kx . Since for each x ∈ E+ it holds

|T |x = sup{|Ty| : y ∈ E, |y| ≤ x}

by [6, Theorem 83.6] in the real case, and by [6, Theorem 92.6] in the complexcase, we conclude that |T |x ≤ Kx for all x ∈ E+ , that is, |T | ≤ K . The proofis now complete.

A collection C of linear operators on E is said to be (absolutely) dom-inated by a positive operator K on E whenever each T ∈ C is (absolutely)dominated by K .

Theorem 3.2. Let E be a Dedekind complete Riesz space, and let M be amaximal semigroup of operators of Lb(E) among those absolutely dominated bya positive operator K . Then

S := {λS : λ ∈ R, S ∈M}

is a subalgebra of Lb(E) and there exists a norm ‖ · ‖ on S such that


(i) ‖ST‖ ≤ ‖S‖ ‖T‖ for all S, T ∈ S ;

(ii) M = {S ∈ S: ‖S‖ ≤ 1} .

Proof. We first show that M is a convex subset of Lb(E). It is easy to seethat the convex hull

co(M) =

{n∑i=1

λiSi : Si ∈M, λi ≥ 0, and

n∑i=1

λi = 1

}

is a semigroup absolutely dominated by K . Since M⊆ co(M), maximality ofM implies that M = co(M), and so M is a convex set. Since M is balancedas well (that is, S ∈ M and |λ| ≤ 1 imply that λS ∈ M), it is not difficult toshow that the semigroup S is a subspace of Lb(E).

It follows that the Minkowski functional

‖S‖ := inf{λ > 0 : S ∈ λM} , S ∈ S

is a seminorm on S and M⊆ C , where

C = {S ∈ S : ‖S‖ ≤ 1}.

If ‖S‖ = 0, then there exists a sequence λn ↓ 0 such that |S| ≤ λnK for all n .Since Lb(E) is Dedekind complete, and hence Archimedean, we obtain |S| = 0.Hence ‖ · ‖ is actually a norm on S .

For the proof of (i) let S and T be operators of S . With no lossof generality we may assume that ‖S‖ = ‖T‖ = 1. Then there exists asequence λn ↓ 1 such that S ∈ λnM and T ∈ λnM for all n . It followsthat ST ∈ (λn)2M for all n , so that ‖ST‖ ≤ 1.

To prove (ii), let S ∈ C . Then there exists a sequence λn ↓ 1 such that|S| ≤ λnK . Since Lb(E) is Archimedean, we have |S| ≤ K . This shows thatthe semigroup C is absolutely dominated by K . Since M ⊆ C , M = C bymaximality of M , and so (ii) holds. The proof is therefore complete.

Let u ∈ E+ , and let f ∈ (E )+ . Then an operator T ∈ Lb(E) isdominated by the operator u ⊗ f if and only if |φ(Tx)| ≤ |φ|(u) f(|x|) for allx ∈ E and φ ∈ E˜. This means that T is dominated by the pair (n, n′)(see Definition 2.2), where the homogeneous functions n: E → [0,∞) andn′: E˜→ [0,∞) are defined by

n(x) = f(|x|) and n′(φ) = |φ|(u).

Furthermore, the following proposition holds for the functions n and n′ .

Proposition 3.3. Assume that 0 6= u ∈ E+ and 0 6= f ∈ (E )+ . Then thehomogeneous functions n: E → [0,∞) and n′: E˜→ [0,∞) defined by

n(x) = f(|x|) and n′(φ) = |φ|(u)


are weak functions on E and E˜ respectively. If u is a strong order unit of E ,then n′ is a strong function on E˜. If f is a strong order unit of E˜, then nis a strong function on E .

Proof. Note that |f(x)| ≤ f(|x|) for all x ∈ E and |φ(u)| ≤ |φ|(u) for allφ ∈ E˜. Therefore, the first assertion is shown by taking ψ := f and y := u inthe definition of weak functions.

Assume that u is a strong order unit of E , and put U := {y ∈ E : |y| ≤u} . Then

|φ(y)| ≤ |φ|(|y|) ≤ |φ|(u) = n′(φ)

for all y ∈ U . Fix x ∈ U . Since u is a strong order unit, there is t > 0 suchthat |x| ≤ t u . Setting λ := 1/t we then have λx ∈ U . This proves that U isan absorbing set and completes the proof of the second assertion.

We omit the proof of the last assertion, since it is similar to the previousone.

Since for each T ∈ Lb(E) the order dual E˜ is invariant under T ′ , wecan apply Theorem 2.3, Theorem 2.8 and Theorem 2.9 in the case Φ := E˜ andLΦ(E) := Lb(E). They together with Proposition 3.3 give the following results.

Theorem 3.4. Assume that 0 6= u ∈ E+ and 0 6= f ∈ (E )+ . Let M be amaximal semigroup of order bounded operators on E among those dominated bythe operator u⊗ f . Then there exist a seminorm ‖ · ‖′1 on E˜ and a seminorm‖ · ‖2 on E such that

(i) ‖x‖2 ≤ ‖x‖1 for all x ∈ E , where

‖x‖1 := sup{|φ(x)| : φ ∈ E , ‖φ‖′1 ≤ 1}

is a quasinorm on E ;

(ii) M = {T ∈ Lb(E): |||T ||| ≤ 1} , where

|||T ||| := sup{‖Tx‖1 : ‖x‖2 ≤ 1}.

If some non-zero multiple of the identity operator I on X is dominated byu⊗ f , then we can achieve that ‖ · ‖1 is a seminorm which is equivalent to theseminorm ‖·‖2 . If I is dominated by u⊗f , then we can even take ‖x‖1 = ‖x‖2for all x ∈ E . Furthermore, if u is a strong order unit of E , then ‖ · ‖′1 canbe chosen to be a norm. Similarly, if f is a strong order unit of E˜ and E˜separates points of E , then ‖ · ‖2 can be chosen to be a norm.

Theorem 3.5. For each i from a given index set I , let 0 6= ui ∈ E+ and0 6= fi ∈ (E )+ . Assume that M is a maximal subsemigroup of Lb(E) among


those dominated by ui ⊗ fi for all i ∈ I . Then for each i ∈ I there exists anoperator quasinorm ||| · |||i on Lb(E) such that

M = {T ∈ Lb(E) : ‖T‖ ≤ 1},

where ‖T‖ := sup{|||T |||i : i ∈ I} .

One may apply the results of this section to the Riesz space lp for1 ≤ p ≤ ∞ . The order and the lattice operations are introduced componentwise.The most interesting operators on this space are infinite matrices. The fact thatthe operator represented by the matrix T = [tij ] is absolutely dominated bythe one represented by the matrix K = [kij ] simply means that |tij | ≤ kij forall indeces i, j .

This example can be extended to the Riesz space L which is an orderideal of M(X,µ), the Riesz space of all equivalence classes of almost equalreal or complex measurable functions. Here µ is a σ -finite measure on a non-empty set X . For an absolute kernel operator T on L and for a positive kerneloperator K on L with respective kernels t(x, y) and k(x, y), we have that Tis absolutely dominated by K whenever |t(x, y)| ≤ k(x, y) almost everywhere,since the kernel of |T | is equal |t(x, y)| by [6, Theorem 94.3].

On the other hand, the above example can be specialized to the case offinite dimensional Riesz space Rn or Cn . If we choose in this case vectors uand f to be respectively the column and the row with all entries equal to 1,then the main conclusion of Theorem 3.3 in [2] becomes a simple consequenceof our Theorem 3.4.

Our results also give a possible answer to the question (implicitly posedin [2]) about characterization of maximal semigroups of matrices dominated bya given positive matrix.

Theorem 3.6. Let K be a matrix of order n with positive entries, and letM be a maximal semigroup of real (or complex) matrices of order n amongthose dominated by K . Then there exist a positive integer r ≤ n and operatorquasinorms ||| · |||i (i = 1, 2, . . . , r ) on Rn×n (or Cn×n ) such that M is equalto the semigroup of all contractions with respect to the norm

‖T‖ := max{|||T |||i : 1 ≤ i ≤ r}.

Moreover, for each i = 1, 2, . . . , r the operator quasinorm |||·|||i is a norm inducedby equivalent norms. If, in addition, I is dominated by K , then these normscan be chosen to be equal.

Proof. We claim that there exist r ≤ n , positive vectors u1 , u2 , . . . , urand positive linear functionals f1 , f2 . . . , fr such that

K = inf{u1 ⊗ f1, . . . , ur ⊗ fr}.


Namely, this can be always achieved with r = n if ui is the i-th column of K ,the i-th entry of fi is 1, and the other entries of fi are large enough. We thenapply Theorems 3.5 and 3.4.

Following [2] we denote by J the matrix of order n with all entries equal1. Furthermore, for a non-negative matrix K = [kij ]

ni,j=1 we denote by SK

the vector space of all matrices A = [aij ]ni,j=1 such that aij = 0 whenever

kij = 0. We conclude the paper by the following extension of Theorem 5.1in [2].

Theorem 3.7. Let K be a non-zero non-negative matrix of order n suchthat SK is a semigroup. Let M be a maximal semigroup of real (or complex)matrices of order n among those dominated by K . Then, as in Theorem 3.6,there exists a norm ‖·‖ on Rn×n (or Cn×n ) such that M = C∩SK , where C isthe semigroup of all contractions with respect to ‖ · ‖ . Moreover, as in Theorem3.6, the same assertions for the operator quasinorms {||| · |||i : 1 ≤ i ≤ r}hold.

Proof. Let λ > 0 be the minimum of all non-zero entries of K . By Theorem3.6 there exists a norm ‖ · ‖ on Rn×n (or Cn×n ) such that the semigroup Cof all contractions with respect to ‖ · ‖ contains M and it is maximal amongthose dominated by sup{K,λJ} . Since M⊆ C∩SK and C∩SK is a semigroupdominated by K , it follows that M = C ∩ SK by maximality of M .

Acknowledgment

The authors acknowledge the support of the Ministry of Science and Technologyof Slovenia.

References

[1] Aliprantis, C. D. and O. Burkinshaw, “Positive Operators”, Academic Press,London, 1985.

[2] Kosir, T., M. Omladic and H. Radjavi, Maximal semigroups dominated by0-1 matrices, Semigroup Forum 54 (1997), 175–189.

[3] Luxemburg, W. A. J. and A. C. Zaanen, “Riesz Spaces I”, North Holland,Amsterdam, 1971.

[4] Meyer-Nieberg, P., “Banach Lattices”, Springer-Verlag, Berlin Heidelberg,1991.


[5] Schaefer, H. H., “Banach Lattices and Positive Operators”, Springer, BerlinHeidelberg New York, 1974.

[6] Zaanen, A. C., “Riesz Spaces II”, North Holland, Amsterdam, 1983.

Faculty of Mathematics and PhysicsUniversity of LjubljanaJadranska 19SI-1000 Ljubljana, [email protected]@fmf.uni-lj.si

Received November 20, 1998Online publication March 15, 2002

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Maximal dominated operator semigroups