On Riesz spaces with b-property and b-weakly compact operators

Riesz spaces considered in this note are assumed to have separating order duals. The order dual of a Riesz space E is denoted by E ^∼ . E ^∼∼ will denote the order bidual of E. The order continuous dual of E is denoted by E _n ^∼ , while E ⁰ will denote the topological dual of a topological Riesz space. E ₊ will denote the cone of positive elements of E . The letters E, F will denote Banach lattices, X, Y will denote Banach spaces. B X will denote the closed unit ball of X . We use without further explanation the basic terminology and results from the theory of Riesz spaces as set out in [1, 2, 14, 17].

Let E be a Riesz subspace of a Riesz space F . A subset of E which is order bounded in F is said to be b-bounded in E. If every b-bounded subset of E remains to be order bounded in E then E is said to have b-property in F . If a Riesz space E has b-property in its order bidual E _n ^∼ then it is said to have b-property.

Riesz spaces with b-property were introduced in [3] and studied in [3–6].

A normed Riesz space E has the weak Fatou property for directed sets if every norm bounded upwards directed set of positive elements in E has a supremum. Riesz spaces with weak Fatou Property for directed sets have b-property. If a Banach lattice has order continuous norm then it has the weak Fatou property for directed sets if and only if it has the b-property [6]. A locally solid Riesz space is said to have Levi property if every topologically bounded set in E ₊ has a supremum. If E is a Frechet lattice with Levi property then E has the b -property [6]. If E is a Dedekind complete locally solid Riesz space with E ⁰ = E ^∼ then E has b -property if and only if E has the Levi property [6]. Thus a Dedekind complete Frechet lattice has Levi property if and only if it has the b -property.

Let E be a Riesz subspace of a Riesz space F . If E is the range of a positive pro jection defined on F then E has b -property in F . If E is a Banach lattice then every sublattice of E isomorphic to li has b-property in E [14, Proposition 2.3.11]. Similarly if the norm of E is order continuous then every sublattice Riesz isomorphic to co has b-property in E [14, Proposition 2.4.3].

Further examples of Riesz spaces with b -property are given in the following example.

(° 2009 Alpay §., Altin B.

Example. A Banach lattice E is called a KB-space if every increasing norm bounded sequence in E ₊ is norm convergent. KB-spaces have b-property. Perfect Riesz spaces have b-property and hence, every order dual has b-property [4]. If K is a compact Hausdorff space and C(K ) is the Riesz space of real valued continuous functions on K under pointwise order and algebraic operations then C(K ) has b-property[4]. On the other hand co real sequences which converge to zero does not have b-property.

An element e > 0 in a Riesz space E is called discrete if the ideal generated by e coincides with the subspace generated by e. A Riesz space E is called discrete if and only if there exists a discrete element v with 0 < v < e for every 0 < e in E .

Example. Discrete elements give rise to ideals with b-property in a Riesz space E. Because if x is a discrete element then the principal ideal I _x generated by x is pro jection band in E and therefore I _x has b-property in E.

T : E ^ F is called b-bounded if T maps b-order bounded subsets of E into b-bounded subsets of F.

T : E ^ X is called b-weakly compact if T maps b-order bounded subsets of E into relatively weakly compact subsets of X .

Although the authors were not aware of this fact until quite recently, much later then the Bolu meeting in fact, b-weakly compact operators were introduced in [15] for the first time under a different name.These operators were studied in [4–11] and in [13–15]. Among b-weakly compact operators T : E ^ X those that map the band B generated by E in E ⁰⁰ into X are called strong type B in [15]. To describe the operators of strong type B, we refer the reader to [13].

A continuous operator T : E ^ X is called order weakly (o-weakly ) compact whenever T [0, x] is a relatively weakly compact subset of X for each x G E+.

A continuous operator T : E ^ X is called AM -compact if T [ — x, x] is relatively norm compact in X for each x G E ₊ .

A continuous operator T from a Banach lattice E into a Banach lattice F is called semicompact if for every e > 0, there exists some u G E ₊ such that T ( B e ) C [ — u, u] + cB f .

A continuous operator T : X ^ Y is called a Dunford-Pettis operator if x n ^ 0 in a(X, X 0 ) implies lim n ||T(x _n ) | = 0.

A b-weakly compact operator is continuous and if W(E, X ) is the space of weakly compact, W b (E, X) is the space of b-weakly compact and Wo(E, X) is the space of order weakly compact operators we have the following relations between these classes of operators:

W(E,X ) C W b (E,X ) C Wo(E,X).

The inclusions may be proper. The identity on L ¹ [0,1] is b-weakly compact but not weakly compact. The identity on co is o-weakly compact but not a b-weakly operator.

If E is an AM -space then W(E,X ) = W _b (E,X). On the other hand Theorem 2.2. in [10] shows that if E ⁰ is a KB -space or X is reflexive then W(E, X ) = W b (E, X). A Banach lattice E is a KB-space if and only if L(E,X ) = W b (E,X) for each Banach space X [5]. If F is a KB -space then again L(E, F) = W b (E,F ) for each Banach lattice E [5]. To generalize, we know that if a Banach space X does not contain c o , then L ( E, X ) = W b ( E, X ).

We need the following characterization of b -weakly compact operators which is a combination of results in [3, 5].

Proposition 1. Let T : E ^ X be an operator. The following are equivalent:

• 1)    T is b-weakly compact.

• 2)    For each b-bounded disjoint sequence (x _n ) in E ₊ , lim n q T (x) = 0 where q T (x) is the Riesz seminorm defined as sup{||T(y) | : | y | 6 | x |} for each x G E.

• 3)    T (x n ) is norm convergent for each b-bounded increasing sequence (x n ) in E ₊ .

• 4)    For each b-bounded disjoint sequence (x n ) in E, we have lim n ||T(x n ) | = 0.

b-weakly compact operators satisfy the domination property. That is, if 0 6 S 6 T and T is b -weakly compact then S is also b -weakly compact which can be seen from the characterization given in Proposition 1(4).

Main results

A Riesz space E is called σ -laterally complete if the supremum of every disjoint sequence of E + exists in E . A Riesz space that is both σ -laterally and σ -Dedekind complete is called σ -universally complete. There exists a universally complete Riesz space E ^u which contains E as an order dense Riesz subspace. E ^u is called the universal completion of E .

The next result exhibits the relation between b -property and σ -lateral completeness. It is actually Theorem 23.23 in [1]. Restated for our purposes it reads as follows.

Proposition 2. Let E be a σ-Dedekind complete Riesz space. Then E is σ-laterally complete if and only if E has b-property in its universal completion E ^u .

The following is Theorem 23.24 in [1].

Corollary. Let E be a Dedekind complete Riesz space. Then E is universally complete if and only if E has countable b-property in E ^u and has a weak order unit.

C As E is order dense in the universal completion E ^u , E is an order ideal of E ^u by Theorem 2.2 in [1]. Suppose E has b-property in E ^u and has a weak order unit e. Let 0 < u' E E ^u be that arbitrary. As e is also a weak order unit of E ^u , we have 0 6 u ' Л ne f u ' . Since E is an ideal, { u ⁰ ∧ ne } ⊆ E and since E has b -property in E ^u , { u ⁰ ∧ ne } is an order bounded subset of E and therefore u ' E E . Hence E = E ^u . B

Examples in [1] show that Dedekind completeness of E and existence of a weak order unit can not be omitted. Theorem 23.32 in [1] shows that among σ -laterally complete Riesz spaces those admitting a Riesz norm or an order unit are those which are Riesz isomorphic to R ⁿ . Thus if E is σ -Dedekind complete and has countable b -property in E ^u which either has an order unit or admits a Riesz norm then E is isomorphic to R ⁿ .

Each order weakly compact operator T : E ^ X factors over a Banach lattice F with order continuous norm as T = SQ where Q is an almost interval preserving lattice homomorphism which is the quotient map E ^ E/q ^— (0) in fact, F is the completion of E/q ^— (0), where q T (x) is the Riesz seminorm defined as sup {|| T(y) | : | y | 6 | x |} for each x E E and S is the operator mapping the equivalence class [x] in E/q ^-1 (0) to T (x) [14,Theorem 3.4.6]. As b -weakly compact operators are order weakly compact every b -weakly compact operator T : E ^ X has a factorization T = SQ over a Banach lattice with order continuous norm. Let us note that if E has order continuous norm then the factorization can be made over a KB -space as if was shown in [7].

This factorization yields a characterization of b -weakly compact operators among order weakly compact operators.

Proposition 3. Let T : E ^ F. T is b-weakly compact if and only if the quotient map Q : E ^ F is b-weakly compact.

C Let F be the completion of Fo = E/q ^-1 (0) and Q be the quotient map Q : E ^ Fo. Since Q is onto, the corresponding operator Q : E ^ F is an almost interval preserving lattice homomorphism.

Suppose T is b-weakly compact and let (x n ) C E ₊ be an b-order bounded disjoint sequence. In view of ||Q(x _n ) k = q T (x n ), we see that lim n k Q(x _n ) k = 0. Thus Q is b-weakly compact by Proposition 1(4).

On the other hand if Q is b-weakly compact then it is easily seen that SQ is also b-weakly compact for each continuous operator S , and thus T = SQ is b-weakly compact. B

This leads us to recapture a result of [5].

C T has a factorization over a Banach lattice H with order continuous norm as SQ where Q : E ^ H is b-weakly compact and S : H ^ F is continuous. Thus | S | exists. The operator | S | Q is b-weakly compact and 0 6 | T | = | SQ | 6 | S | Q. Thus | T | is a b-weakly compact as b -weakly compact operators satisfy the domination property. B

A deficiency of b -weakly compact operators is that they do not satisfy the duality property. For example, the identity I on li is b-weakly compact but its adjoint, the identity on l ^ , is not b-weakly compact. On the other hand the identity on cg is not b-weakly compact but its adjoint, the identity on li, is certainly b-weakly compact. For recent developments on duality of b -weakly compact operators we refer the reader to [9].

One of the sufficients conditions for an operator to be b -weakly compact is that for each b-bounded disjoint sequence (x n ) in the domain we have lim n | T(x n ) | = 0. Utilizing this it is easy to see that b-weakly compact operators are norm closed in L(E,X). A result in [12] shows that strong limit of o-weakly compact operators is also o-weakly compact under certain conditions. The following example shows that b-weakly compact operators behave differently in this respect.

We will call an operator T : E ^ X summing if T maps weakly summable sequences in E to summable sequences in X.

Proposition 4. Let T : E ^ X be a summing operator between a Banach lattice E and a Banach space X . Then T is b -weakly compact.

C Let (e _n ) be a b-bounded disjoint sequence in E+. It suffices to show that (T(x n )) is norm convergent to 0. There exists an e in E '+ such that 0 6 ^2 e k 6 e for each partial sum. It follows that the sequence (e ^ ) is a weakly summable sequence in E. As T is summing, we have ^Te k <  to , and hence | Te k || ^ 0 in X. B

It is easy to see that an operator T : E ^ X is b-weakly compact if and only if the operator j x T : E ^ X ⁰⁰ is b-weakly compact where j x is the canonical embedding of X into X ⁰⁰ . Let us recall that an operator T : E ^ X is called injective if T is one-to-one and has closed range. Generalizing the previous observation slightly we show that for an operator to be b-weakly compact the size of the target space does not matter.

Proposition 5. Let T : E ^ X and j : X ^ Y be operators where j is an injection. Then T is b -weakly compact if and only if jT is b -weakly compact.

Using the characterization of b-weakly compact operators given in Proposition 1(4) it follows immediately that every Dunford-Pettis operator T : E ^ X is actually a b-weakly compact operator. On the other hand the result in [11] shows that if E has weakly sequentially continuous lattice operations and has an order unit then every positive order weakly compact, in particular every b-weakly compact operator T : E ^ X is a Dunford-Pettis operator. Let us note however that weak sequential continuity of the lattice operations only is not sufficient. Indeed, the identity operator on c0 is o-weakly compact but not a Dunford-Pettis operator although c0 has weakly continuous lattice operations.

In opposite direction we have the following result which is a slight improvement of theorem 2.1 in [11].

Proposition 6. If each positive b-weakly compact operator T : E ^ F is a Dunford-Pettis operator then either E has weakly sequentially continuous lattice operations or F has order continuous norm.

C Let S and T be two operators from E into F satisfying 0 6 S 6 T and T be a Dunford-Pettis operator.Then T is a b -weakly compact operator. As b -weakly compact operators satisfy the domination property S is also a b -weakly compact operator. By the assumption S is a Dunford-Pettis operator. The result now follows from Theorem 3.1 in [16]. B

Now we investigate the relation between b -weakly compact operators and AM -compact operators. The natural embedding j : L “ [0,1] ^ L ^p [0,1], 1 6 p <  ∞ is a b -weakly compact operator which is not AM -compact.

Proposition 7. Let E, F be Banach lattices with E ⁰ discrete. Then every o-weakly compact (and therefore every b-weakly compact) operator from E into F is AM -compact.

C It suffices to show that T [0, x] is relatively norm compact for each x G E+. Let S be the restriction of T to the principal order ideal I x generated by x. Then S : I x ^ F and S 0 : F 0 ^ I x are both weakly compact operators. Therefore S ⁰ ( B f о ) is relatively compact in o(I x , I xX ). I x is an AL-space. Let A be the solid hull of S ⁰ ( B f о ) in I x . Every disjoint sequence in A is convergent for the norm in I _x ⁰ by Theorem 21.10 in [1]. Since E ⁰ is assumed to be discrete, A is contained in the band generated by discrete elements of I _x ⁰ . Employing Theorem 21.15 in [1], we see that A is relatively compact for the norm of I 0 . Therefore S ⁰ : F ⁰ ^ Г _х is a compact operator. Consequently, T : I x ^ F is also compact and thus T [0, x] is relatively compact in F . B

If T : E ^ E is a b-weakly compact operator then T ² is also a b-weakly compact but not necessarily a weakly compact operator. For example the identity I on L ¹ [0,1] is b-weakly compact as L ¹ [0,1] is a KB-space [3], but 1 ² is not a weakly compact operator. It has recently been shown that for a positive b-weakly compact operator T : E ^ E , T ² is weakly compact if and only if each positive b-weakly compact operator T : E ^ E is weakly compact [10, Theorem 2.8].

Now we will now study compactness of b -weakly compact operators.

Proposition 8. Suppose that every positive b-weakly compact operator is compact. Then one of the following holds:

• 1)    E ⁰ and F have order continuous norms.

• 2)    E ⁰ is discrete and has order continuous norm.

• 3)    F is discrete and has order continuous norm.

C Let S, T : E ^ F be such that 0 6 S 6 T where T is compact. Then T and S are b-weakly compact operators. Thus S is compact by the hypothesis. The conclusion now follows from Theorem 2.1 in [16]. B

On the compactness of squares of b -weakly compact operators we have the following. The proof is very similar to the proof of the preceding proposition. Therefore it is omitted.

Proposition 9 . Let E be a Banach lattice with the property that for each positive b-weakly compact operator S : E ^ E, S ² is compact. Then one of the following holds.

• 1)    E has order continuous norm.

• 2)    E ⁰ has order continuous norm.

• 3)    E ⁰ is discrete.

b -property has been very useful in characterizing KB -spaces. For example a Banach lattice E is a KB -space if and only if E has order continuous norm and b -property or if and only if the identity operator on E is b -weakly compact [3–4].

We now present another characterization of KB -spaces.

Proposition 10. A Banach lattice F is a KB-space if and only if for each Banach lattice E and positive disjointness preserving operator T : E ^ F, T is b-weakly compact.

C If the hypothesis on F is true then taking E = F , we see that the identity on E is b-weakly compact and thus E is a KB -space [3]. On the other hand if (x _n ) is a b-bounded disjoint sequence in E+, then (Tx n ) is an order bounded disjoint sequence in F as there exists a positive projection of F ⁰⁰ onto F . Then ||T(x _n ) | ^ 0 as a KB-space has order continuous norm. It follows from Proposition 1(4) that T is b -weakly compact. B

Proposition 11. Consider operators T : E ^ F and S : F ^ G. Suppose S is strong type B and T ⁰ is b-weakly compact. Then ST is a weakly compact operator.

C It suffices to show (ST) ⁰⁰ (E ⁰⁰ ) C G. Since order dual of a Banach lattice has b-property, T ⁰ is o -weakly compact and being so, T has factorization over a Banach lattice H with order continuous dual norm as T = T1T0 where To : E ^ H is continuous and Ti : H ^ F is an interval preserving lattice homomorphism by Theorem 3.5.6 in [14]. Since H ⁰ has order continuous norm, we have (H ⁰ ) ⁰ _n = H ⁰⁰ and T”((H 0 ) П ) C (F ⁰ ) ⁰n as T" is order continuous. Now the weak compactness of ST follows from

(STy\E") = S0(T1(T0(E"W C S"TyH00)) C S00(T10(H0)П) C S00(F0)0n C G where the last inclusion follows from the fact that S is of strong type B and therefore S00 maps the band (F0)'n generated by F in F00 into G. B

As order duals have b -property, assuming T ⁰ to be b -weakly compact is the same as assuming it to be o -weakly compact. Also, we could have taken T to be semicompact as T ⁰ is o -weakly compact whenever T is semicompact [14, Theorem 3.6.18].

Corollary. Let T be an operator on a Banach lattice such that both T and T ⁰ are strong type B. Then T ² is weakly compact.

Finally we study the relationship between semicompact and b -weakly compact operators. It is immediate from the definitions and Theorem 14.17 in [2] that if the range has order continuous norm, thus ensuring weak compactness of order intervals, each semicompact operator is weakly and therefore b -weakly compact.

On the other hand the identity I on l 1 is a b -weakly compact operator which is not semicompact. Theorem 127.4 in [17] shows that if E ⁰ and F have order continuous norms then every order bounded semicompact operator T : E ^ F is b-weakly compact.

The next result gives necessary and sufficient conditions for a Banach lattice to be a KB -space as well as illuminates the relation between semicompact and b -weakly compact operators.

First we need a Lemma which was first proved in [9].

Lemma. Let E be a Banach lattice. If (e n ) is a positive disjoint sequence in E such that ||e n k = 1 for all n, then there exists a positive disjoint sequence (g n ) in E ⁰ with | g_n | 6 1 and satisfying g _n (e _n ) = 1 and g _n (e m ) = 0 for all n = m.

C Let (en) be a disjoint sequence in E+ with |en| = 1 for all n. By Hahn-Banach Theorem there exists fn E E’+ such that |fn| = 1 and fn(en) = |en| = 1. Considering E in (E0)0n, we see that carriers Cen of en are mutually disjoint bands in E0. If gn is the pro jection of fn onto Cen, then the sequence (gn) has the desired properties. B

Let us recall that a Banach lattice E is said to have the Levi Property if every increasing norm bounded net in E ₊ has a supremum in E ₊ . It is well-known that a Banach lattice with Levi Property is Dedekind complete.

The following result gives a necessary and sufficient conditions for a Banach lattice to be a KB-space.

Proposition 12. Let E and F be Banach lattices and assume that F has the Levi property. Then the following are equivalent:

• 1)    Each continuous operator T : E ^ F is b-weakly compact.

• 2)    Each continuous semicompact operator T : E ^ F is b-weakly compact.

• 3)    Each positive semicompact operator T : E ^ F is b-weakly compact.

• 4)    Either E or F is a KB-space.

C It is clear that 1) implies 2) and 2) implies 3). The implication 4) ⇒ 2) was proved in [5]. We will prove that 3) implies 4).

Let us assume that neither E nor F is a KB -space. To finish the proof we construct a positive semicompact operator T : E ^ F which is not b-weakly compact. Recall that a Banach lattice is a KB -space if and only if the identity operator on it is b -weakly compact[3]. Thus if E is not a KB-space, there exists a b-bounded disjoint sequence (e _n ) in E ₊ with k e n k = 1 for all n. Hence by the Lemma, there exists a positive disjoint sequence (g n ) in E 0 with ||g n k 6 1 such that g _n (e _n ) = 1, g n (e _m ) = 0 for all n = m.

We define a positive operator Ti : E ^ l ^ as follows:

x ^ Ti(x) = (gi(x), g2(x),...)

for each x in E . Let us note that T i ( B e ) C B i ^ .

On the other hand, since F is not a KB-space, we can find a b-bounded disjoint sequence in F+ such that 0 6 fn 6 f for some f in F00 and satisfying |fn| = 1 for all n. Let (an) be a positive sequence in l∞ . Then, n        n+1

• 0 6 X a i f i 6 X a i f i 6 sup(a i )f i =1          i =1

shows that the sequence (522 =1 a i f i^n is an increasing norm bounded sequence in F . As F is assumed to have the Levi Property, supremum of (522=₁ a i f i^n exists in F . We denote this supremum by 522=1 a i f i . This enables us to define an operator T2 : l ⁺ ^ F by T2(a i ) = P2=i a i f i .

T2 has an extension to l ₂ which we will also denote by T2.

Since ( f i ) is a disjoint sequence, it follows from

n

0 6    fi =   fi 6 f i=1

that 0 6 ( 522 =1 f i ) n is also an increasing norm bounded sequence in F+. Therefore the supremum of this sequence exists in F and will be denoted by fo- Then T ₂ (B /^ ) C [ — fo, fo]. Now we consider the operator T = T 2 T 1 defined as

∞ x ^ ^g i ^{( x ) f} i i =1

T is well-defined and is positive. It follows from

T ( B e ) = T 2 T i ( B e ) C T 2 B l ) C [ - fo, fo]

that T is semicompact. However, the operator T is not b-weakly compact as

∞

T (en) = X 9i(en)fi = fn i=1

for all n and ||T(e _n ) k = k f n k = 1 for all n. Recall that if T were b-weakly compact then we would have T(e n ) ^ 0 in norm. B

The assumption that F has Levi Property is essential. In fact, if we take E = l _ro , F = co, then each operator from E into F is weakly compact and therefore b -weakly compact. However neither E nor F is a KB -space.