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Vladikavkaz Mathematical Journal 2009, Vol. 11, No 2, pp. 19-26
UDC 517.98
ON RIESZ SPACES WITH b-PROPERTY AND b-WEAKLY COMPACT OPERATORS
§. Alpay, B. Altin
An operator T : E ^ X between a Banach lattice E and a Banach space X is called b-weakly compact if T(B) is relatively weakly compact for each b-bounded set B in E. We characterize b-weakly compact operators among o-weakly compact operators. We show summing operators are b-weakly compact and discuss relation between Dunford-Pettis and b-weakly compact operators. We give necessary conditions for b-weakly compact operators to be compact and give characterizations of KB-spaces in terms of b-weakly compact operators defined on them.
Mathematics Subject Classification (2000): 46A40, 46B40, 46B42. Key words: b-bounded sets, b-weakly compact operator, KB-spaces.
Introduction
Riesz spaces considered in this note are assumed to have separating order duals. The order dual of a Riesz space E is denoted by EEwill denote the order bidual of E. The order continuous dual of E is denoted by E~, while E' 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. Bx 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~ 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' = 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 projection 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 Ix generated by x is projection band in E and therefore Ix 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'' 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(Be) Q [—u, u] + eB^.
A continuous operator T : X ^ Y is called a Dunford-Pettis operator if xn ^ 0 in a(X, X') implies limn ||T(xn)|| = 0.
A b-weakly compact operator is continuous and if W(E, X) is the space of weakly compact, Wb(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) Q Wb (E, X) Q Wo(E,X).
The inclusions may be proper. The identity on L1 [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) = Wb(E, X). On the other hand Theorem 2.2. in [10] shows that if E' is a KB-space or X is reflexive then W(E, X) = Wb(E, X). A Banach lattice E is a KB-space if and only if L(E, X) = Wb(E,X) for each Banach space X [5]. If F is a KB-space then again L(E, F) = Wb(E,F) for each Banach lattice E [5]. To generalize, we know that if a Banach space X does not contain Co, then L(E, X) = Wb(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 (xn) in E+, limn qr(x) = 0 where qr(x) is the Riesz seminorm defined as sup{||T(y)|| : |y| ^ |x|} for each x G E.
3) T(xn) is norm convergent for each b-bounded increasing sequence (xn) in E+.
4) For each b-bounded disjoint sequence (xn) in E, we have limn ||T(xn)|| = 0.
b-weakly compact operators satisfy the domination property. That is, if 0 ^ S ^ 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 a-laterally complete if the supremum of every disjoint sequence of E+ exists in E .A Riesz space that is both a-laterally and a-Dedekind complete is called a-universally complete. There exists a universally complete Riesz space Eu which contains E as an order dense Riesz subspace. Eu is called the universal completion of E.
The next result exhibits the relation between b-property and a-lateral completeness. It is actually Theorem 23.23 in [1]. Restated for our purposes it reads as follows.
Proposition 2. Let E be a a-Dedekind complete Riesz space. Then E is a-laterally complete if and only if E has b-property in its universal completion Eu.
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 Eu and has a weak order unit.
< As E is order dense in the universal completion Eu, E is an order ideal of Eu by Theorem 2.2 in [1]. Suppose E has b-property in Eu and has a weak order unit e. Let 0 < u' G Eu be that arbitrary. As e is also a weak order unit of Eu, we have 0 ^ u' A ne | u'. Since E is an ideal, {u' A ne} C E and since E has b-property in Eu, {u' A ne} is an order bounded subset of E and therefore u' G E. Hence E = Eu. >
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 a-laterally complete Riesz spaces those admitting a Riesz norm or an order unit are those which are Riesz isomorphic to . Thus if E is a-Dedekind complete and has countable b-property in Eu which either has an order unit or admits a Riesz norm then E is isomorphic to
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—1(0) in fact, F is the completion of E/q—1(0), where qT(x) is the Riesz seminorm defined as sup{||T(y)|| : |y| ^ |x|} for each x G 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.
< 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 (xn) C E+ be an b-order bounded disjoint sequence. In view of ||Q(xn)|| = qT(xn), we see that limn ||Q(xn)|| = 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. >
This leads us to recapture a result of [5].
Corollary. Suppose that T : E ^ F is b-weakly compact where F is a Dedekind complete AM-space with order unit.Then |T| is a b-weakly compact operator.
< 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 ^ |T| = |SQ| ^ |S|Q. Thus |T| is a b-weakly compact as b-weakly compact operators satisfy the domination property. >
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 co 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 (xn) in the domain we have limn ||T(xn)|| = 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.
Example.For each n, let Tn : co ^ co be defined as Tn(y) = (yi,..., yn, 0,...). Then the finite rank operators (Tn) are b-weakly compact for each n and we have Tn(y) ^ I(y) for each y in co. However the identity operator I on co is not a b-weakly compact operator.
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.
< Let (en) be a b-bounded disjoint sequence in E+. It suffices to show that (T(xn)) is norm convergent to 0. There exists an e in E+ such that 0 ^ ^ ek ^ e for each partial sum. It follows that the sequence (ek) is a weakly summable sequence in E. As T is summing, we have ^Tek < to, and hence ||Tek|| ^ 0 in X. >
It is easy to see that an operator T : E ^ X is b-weakly compact if and only if the operator jxT : E ^ X" is b-weakly compact where jx is the canonical embedding of X into X". 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 Co is o-weakly compact but not a Dunford-Pettis operator although Co 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.
< Let S and T be two operators from E into F satisfying 0 ^ S ^ 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]. >
Now we investigate the relation between b-weakly compact operators and AM-compact operators. The natural embedding j : L^[0,1] — Lp[0,1], 1 ^ p < to is a b-weakly compact operator which is not AM-compact.
Proposition 7. Let E, F be Banach lattices with E' discrete. Then every o-weakly compact (and therefore every b-weakly compact) operator from E into F is AM-compact.
< 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 Ix generated by x. Then S : Ix — F and S' : F' —> I' are both weakly compact operators. Therefore S'(Bp/) is relatively compact in a(Ix, I''). I' is an AL-space. Let A be the solid hull of S'(Bp/) in I'. Every disjoint sequence in A is convergent for the norm in I' by Theorem 21.10 in [1]. Since E' is assumed to be discrete, A is contained in the band generated by discrete elements of I ' . Employing Theorem 21.15 in [1], we see that A is relatively compact for the norm of I'. Therefore S' : F' — I ' is a compact operator. Consequently, T : Ix — F is also compact and thus T[0, x] is relatively compact in F. >
If T : E — E is a b-weakly compact operator then T2 is also a b-weakly compact but not necessarily a weakly compact operator. For example the identity I on L1 [0,1] is b-weakly compact as L1[0,1] is a KB-space [3], but 12 is not a weakly compact operator. It has recently been shown that for a positive b-weakly compact operator T : E — E, T2 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' and F have order continuous norms.
2) E' is discrete and has order continuous norm.
3) F is discrete and has order continuous norm.
< Let S, T : E — F be such that 0 ^ S ^ 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]. >
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, S2 is compact. Then one of the following holds.
1) E has order continuous norm.
2) E' has order continuous norm.
3) E' 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.
< 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 (xn) is a b-bounded disjoint sequence in E+, then (Txn) is an order bounded disjoint sequence in F as there exists a positive projection of F'' onto F. Then ||T(xn)|| ^ 0 as a KB-space has order continuous norm. It follows from Proposition 1(4) that T is b-weakly compact. >
Proposition 11. Consider operators T : E ^ F and S : F ^ G. Suppose S is strong type B and T' is b-weakly compact. Then ST is a weakly compact operator.
< It suffices to show (ST)''(E'') C G. Since order dual of a Banach lattice has b-property, T' is o-weakly compact and being so, T has factorization over a Banach lattice H with order continuous dual norm as T = TiTo 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' has order continuous norm, we have (H')n = H'' and Ti'((H')^) C (F')n as Ti' is order continuous. Now the weak compactness of ST follows from
(ST)''(E'') = S''(T1''(T0'(E''))) C S''(Ti'(H'')) C S''(T/'(H')n) C S''(F')n C G
where the last inclusion follows from the fact that S is of strong type B and therefore S'' maps the band (F')n generated by F in F'' into G. >
As order duals have b-property, assuming T' 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' 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' are strong type B. Then T2 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 li is a b-weakly compact operator which is not semicompact. Theorem 127.4 in [17] shows that if E' 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 (en) is a positive disjoint sequence in E such that ||en|| = 1 for all n, then there exists a positive disjoint sequence (gn) in E' with ||gn|| ^ 1 and satisfying gn(en) = 1 and gn(em) = 0 for all n = m.
< Let (en) be a disjoint sequence in E+ with ||en|| = 1 for all n. By Hahn-Banach Theorem there exists fn £ E+ such that ||fn|| = 1 and fn (en) = ||en || = 1. Considering E in (E')n, we
see that carriers Cen of en are mutually disjoint bands in E'. If gn is the projection of fn onto Cen, then the sequence (gn) has the desired properties. >
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.
< 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 (en) in E+ with ||en|| = 1 for all n. Hence by the Lemma, there exists a positive disjoint sequence (gn) in E' with ||gn|| ^ 1 such that gn(en) = 1, gn(em) = 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 Ti(Be) Q B^.
On the other hand, since F is not a KB-space, we can find a b-bounded disjoint sequence in F+ such that 0 ^ fn ^ f for some f in F'' and satisfying ||fn|| = 1 for all n. Let (an) be a positive sequence in l+. Then,
n n+1
0 < af ^ af ^ sup(a»)f i=1 i=1
shows that the sequence (^n=i af)n is an increasing norm bounded sequence in F. As F is assumed to have the Levi Property, supremum of (^n=i af)n exists in F. We denote this supremum by ^ °=i a» f. This enables us to define an operator T2 : 1+ —> f by T2(a) =
T2 has an extension to l+ which we will also denote by T2. Since (fj) is a disjoint sequence, it follows from
n
0 f = V f < f i=i
that 0 ^ (^n=i fi)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 T2(B;tc ) Q [—fo, fo].
Now we consider the operator T = T2Ti defined as
+
x — Y1 gj(x)fj i=i
T is well-defined and is positive. It follows from
T(Be) = T2T1 (BE) C T^) C [-/0, /0 ] that T is semicompact. However, the operator T is not b-weakly compact as
T(en) = ^gi(en)fi = i=1
for all n and ||T(en)|| = ||/n|| = 1 for all n. Recall that if T were b-weakly compact then we would have T(en) ^ 0 in norm. >
The assumption that F has Levi Property is essential. In fact, if we take E = 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.
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Received February 4, 2009. §afak Alpay
Department of Mathematics
Middle East Technical University, Prof. Dr.
Turkiye, Ankara
E-mail: safak@metu.edu.tr
Biröl Altin
Department of Mathematics Gazi Universitesi, Asc. Prof. Dr. Turkiye, Besevler-Ankara E-mail: birola@gazi.edu.tr
