Abstract
The main aim of this paper is studying the family \(W_b(E,F)\) of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if \(T:G\rightarrow F\) is a b-weakly compact operator between two Banach lattices, then \(T\in W_b(E,F)\) whenever the norm of E is order continuous and \(T:E\rightarrow F\) is a positive operator. We also investigate the relationship between \(W_b(E,F)\) and some other classes of operators like \(L^{(1)}_c(E,F)\) and \(L^{(2)}_c(E,F)\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Preliminaries
An operator T from a Banach lattice E to a Banach space X is said to be b-weakly compact, if the image of every b-order bounded subset of E (that is, order bounded in the topological bidual \(E^{\prime \prime }\) of E) under T is relatively weakly compact. The authors in [8] proved that an operator T from a Banach lattice E into a Banach space X is b-weakly compact if and only if \(\{Tx_n\}_n\) is norm convergent for every positive increasing sequence \(\{x_n\}_n\) of the closed unit ball \(B_E\) of E. The class of all b-weakly compact operators between E and X will be denoted by \(W_b(E,X)\). The class of b-weakly compact operators was firstly introduced by Alpay et al. [3]. One of the interesting properties of the class of b-weakly compact operators is that it satisfies the domination property. Some more investigations on \(W_b(E,X)\) were done by [3,4,5, 7, 8].
In this paper, we continue the investigation on \(W_b(E,X)\). In the first section, we provide some prerequisites. The second section is devoted to the main results. We mainly focus on the inclusion relationship between \(W_b(E,X)\) with some known class of operators. We also study those Banach lattices for which the modulus of an order bounded operator is b-weakly compact.
1.1 Some Basic Definitions
Let E be a vector lattice. An element \(e>0\) in E is said to be an order unit whenever for each \(x\in E\) there exists a \(\lambda >0\) with \(|x| \le \lambda e\). A sequence \((x_n)\) in a vector lattice is said to be disjoint whenever \(|x_n| \wedge |x_m| =0\) holds for \(n\ne m\). A vector lattice is called Dedekind complete whenever every nonempty bounded above subset has a supremum. For an operator \(T:E\rightarrow F\) between two vector lattices, we shall say that its modulus |T| exists whenever \(|T|:=T\vee (-T)\) exists in the sense that |T| is the supremum of the set \(\{-T,T\}\). An operator \(T:E\rightarrow F\) between two vector lattices is called order bounded if it maps order bounded subsets of E into order bounded subsets of F. An operator \(T: E\rightarrow F\) is said to be positive if \(T(x)\ge 0\) in F whenever \(x\ge 0\) in E. Note that each positive linear operator on a Banach lattice is continuous and order bounded. A Banach lattice E has order continuous norm if \(\Vert x_\alpha \Vert \rightarrow 0\) for every decreasing net \((x_\alpha )\) with \(\inf _\alpha x_\alpha =0\). If E is a Banach lattice, its topological dual \(E^\prime \), endowed with the dual norm and dual order, is also a Banach lattice. A Banach lattice E is said to be an AM-space if for each \(x,y\in E^+\) such that \(x\wedge y=0\), we have \(\Vert x\vee y\Vert = max (\Vert x\Vert , \Vert y\Vert )\). A Banach lattice E is said to be a KB-space whenever each increasing norm bounded sequence of \(E^+\) is norm convergent. It is known that every reflexive Banach lattice is a KB-spaces. Moreover, each KB-space has order continuous norm. Recall that an operator T from a Banach lattice E into a Banach space X is said to be order weakly compact, if it maps each order bounded subset of E into a relatively weakly compact subset of X, i.e., \(T[-x,x]\) is relatively weakly compact in X for each \(x\in E^+\).
A positive linear operator \(T:E\rightarrow F\) is called almost interval preserving if T[0, x] is dense in [0, Tx] for every \(x\in E^+\). Let E be a vector lattice. A sequence \((x_n)\subset E\) is called order convergent to x if there exists a sequence \((y_n)\) such that \(y_n \downarrow 0\) and for some \(n_0\), \(|x_n -x| \le y_n\) for all \(n\ge n_0\). We will write \(x_n\xrightarrow {o_1}x\) when \((x_n)\) is order convergent to x. A sequence \((x_n)\) in a vector lattice E is strongly order convergent to \(x\in E\), denoted by \(x _n \xrightarrow {o_2}x\) whenever there exists a net \((y_\beta )\) in E such that \(y_\beta \downarrow 0\) and that for every \(\beta \), there exists a \(n_0\) such that \(|x_n -x| \le y_\beta \) for every \(n\ge n_0\). It is clear that every order convergent sequence is strongly order convergent.
For more information concerning the Banach lattice and the related topics, we refer the reader to [2, 10].
2 Results
We commence with the following result showing that if an operator is b-weakly compact on some order dense sublattice, then it will be b-weakly compact on the whole space.
Theorem 2.1
Let E and F be two Banach lattices such that the norm of E is order continuous and let \(T:E\rightarrow F\) be a positive operator. Then, for an order dense sublattice G of E, if \(T\in W_b(G,F)\) then \(T\in W_b(E,F)\).
Proof
Let \((x_n)\) be a bounded positive increasing sequence in E. Since G is order dense in E, from [2, Theorem 1.34] we have \(\{y\in G:~0\le y\le x_n\}\uparrow x_n,\) for every n. Let \((y_{mn})_m\subset G\) with \(0\le y_{mn}\uparrow x_n\) for every n and set \(z_{mn}=\vee _{i=1}^n y_{mi}\). It follows that \(z_{mn}\uparrow _m x_n\) and \(\sup _{m,n}\Vert z_{mn}\Vert \le \sup _n\Vert x_n\Vert <\infty \). Now, if \(T\in W_b(G,F)\), then \((Tz_{mn})\) is norm convergent to some \(y\in F\). Then, from
we get
and this completes the proof. \(\square \)
Definition 2.2
Let E and F be two vector lattices. We define \(L^{(1)}_c(E,F)\) (resp. \(L^{(2)}_c(E,F)\)) as the collection of all order bounded operators T for which \(x_n\xrightarrow {o_1}0\) (resp. \(x_n\xrightarrow {o_2}0\)) implies \(Tx_{n_k} \xrightarrow {o_1}0\) (resp. \(Tx_{n_k} \xrightarrow {o_2}0\)) for some subsequence \((x_{n_k})\) of \((x_{n})\).
It should be noted that \(L^{(1)}_c(E,F)=L^{(2)}_c(E,F)\) when F is Dedekind complete, see for example [1], in which there are also examples showing that these two collections can be different.
Theorem 2.3
Let E and F be two Banach lattices such that the norm of E is order continuous. Then
-
(1)
\(W_b(E,F)^+ \subseteq L^{(2)}_c(E,F)\).
-
(2)
If \(W_b(E,F)\) is a vector lattice and F is Dedekind complete, then \(W_b(E,F)\) is an order ideal of \( L^{(1)}_c(E,F)= L^{(2)}_c(E,F)\).
Proof
-
(1)
Let \(T\in W_b(E,F)^+\) and let \((x_n)\subset E\) be a strongly order convergent sequence. Without lose of generality, we assume that \(0\le x_n\xrightarrow {o_2} 0\), which follows that \((x_n)\) is norm convergent to zero. Set \((x_{n_j})\) as a subsequence with \(\sum _{j=1}^{\infty }\Vert x_{n_j} \Vert <\infty \). Define \(y_m=\sum _{j=1}^m x_{n_j}\). Then, \((y_m)\) is bounded and \(0\le y_m\uparrow \). Since T is a b-weakly compact operator, \((Ty_m)\) is norm convergent to some point \(z\in F\). Using [9, Lemma 3.11], \((Ty_m)\) has a subsequence \((Ty_{m_k})\) strongly order convergent to \(z\in F\). Thus, there exists a net \((z_\beta )\subset F^+\) with the property that for each \(\beta \) there exists some \(n_0\) such that if \(k\ge n_0\), then \(\vert Ty_{m_k}-z\vert \le z_\beta \downarrow 0\). Consequently
$$\begin{aligned} 0\le Tx_{n_{m_k}}&\le \vert Ty_{m_k}-Ty_{m_{k'}}\vert \\&\le \vert Ty_{m_k}-z\vert +\vert Ty_{m_{k'}}-z\vert \\&\le z_\beta +z_\beta \downarrow 0, \end{aligned}$$for every \(k\ge k' \ge n_0\), which confirms that \(T\in L^{(2)}_c(E,F)\) as required.
-
(2)
By [2, Corollary 4.10], E is Dedekind complete, so by Dedekind completeness of F, \( L^{(2)}_c(E,F)=L^{(1)}_c(E,F)\). Furthermore, since \(W_b(E,F)\) is a vector lattice, it follows from part (1) that \(W_b(E,F)\) is a subspace of \( L^{(1)}_c(E,F)\). Now proof follows from the fact that \(W_b(E,F)\) satisfies the domination property.
\(\square \)
We need the following elementary lemma in some of the forthcoming results in this section.
Lemma 2.4
Let E and X be Banach spaces and let \(T:E\rightarrow X\) be a bounded linear operator with closed range. Then
-
(1)
If T is compact, then T is of finite rank.
-
(2)
If T is weakly compact, then T(E) is reflexive.
Proof
-
(1)
If T is compact, then T(U) is relatively norm compact, where U is the open unit ball in E. On the other hand, by the open mapping theorem, T(U) is open. It follows that the Banach subspace T(E) of X is locally compact, so it must be of finite dimensional, as claimed.
-
(2)
If T is weakly compact, then T(B) is relatively weakly compact, where B denotes the closed unit ball in E. This fact together with the equality \(T(E)=\bigcup _{n\in {\mathbb {N}}}nT(B)\) implies that the unit closed ball in T(E) is weakly compact. Consequently, T(E) must be reflexive.
\(\square \)
Proposition 2.5
Let E be a Banach lattice and let X be a non-reflexive Banach space. If \(T:E\rightarrow X\) is a surjective b-weakly compact operator, then the norm of \(E^\prime \) is not order continuous.
Proof
If the norm of \(E^\prime \) is order continuous, then by [7, Theorem 2.2] T must be weakly compact and Lemma 2.4 implies that X is reflexive which is a contradiction.
Proposition 2.6
Let E and F be two Banach lattices such that the norm of \(F^\prime \) is order continuous. If \(T:E \rightarrow F\) is an injective almost interval preserving and b-weakly compact operator with closed range, then E is reflexive.
Proof
Since T has closed range, we may assume, without loss of generality, that T is onto. Thus, \(T:E \rightarrow F\) is a bijection between two Banach spaces and it follows that \(T^\prime :F^\prime \rightarrow E^\prime \) is also a bijection. On the other hand, since T is almost interval preserving, by [10, Theorem 1.4.19], \(T^\prime \) is a lattice homomorphism, so by [2, Theorem 2.15], both \(T^\prime \) and \({(T')^{-1}}\) are positive operators. Since the norm of \(F^\prime \) is order continuous, by [2, Theorem 4.59], \(F^\prime \) is a KB-space, so \(T^\prime \) is b-weakly compact and the norm of \(E^\prime \) is also order continuous. Since T is b-weakly compact, by [7, Theorem 2.2], T is weakly compact. It follows that \(T^\prime \) is also weakly compact. Now Lemma 2.4 implies that \(E^\prime \) must be reflexive, so E is reflexive, as claimed.
Proposition 2.7
Let E be a Banach lattice, X be a Banach space and let \(T:E\rightarrow X\) be an injective b-weakly compact operator with closed range. Then, E is finite-dimensional when either of the following conditions hold.
-
(1)
E is an AM-space with order continuous norm.
-
(2)
E is an AM-space and \(E^\prime \) is discrete.
Proof
Similar to the proof of Proposition 2.6, we may assume that T is onto, so \(T^\prime :X^\prime \rightarrow E^\prime \) is a also onto. Then by [8, Proposition 2.3], T is a compact operator under either of the conditions (1) and (2). Thus, \(T^\prime \) is compact. Now, by Lemma 2.4, \(E^\prime \) is finite dimensional. Hence, E is finite dimensional.
Theorem 2.8
For two Banach lattices E and F, if E has order unit and the norm of F is order continuous, then every order bounded operator \(T:E\rightarrow F\) is b-weakly compact operator.
Proof
Let \(T:E\rightarrow F\) be a bounded operator and take a bounded increasing sequence \((x_{n})\) in E. Let \(e \in E^{+}\) be an order unit for E. For each \(x \in E\), the norm \(\Vert x\Vert _{\infty }=\inf \lbrace \lambda >0 : \vert x \vert \le \lambda e\rbrace \) on E is equivalent to the original norm, so it follows that \(\mathop {\text {sup}}_{n} \Vert x_{n}\Vert _{\infty }< \infty \). For each \(n\in {\mathbb {N}} \), there exists a \(\lambda _n>0\) such that \(\lambda _{n} \le \Vert x_{n}\Vert _{\infty }+1\) and \(\vert x_{n}\vert \le \lambda _{n}e\). Then, we get
In particular, \(( x_{n}) \subseteq [- \lambda e , \lambda e]\). Now since F has order continuous norm, it is Dedekind complete, and so \(T^+\) exists. It follows that \(( T^+ x_{n}) \subseteq T^+[-\lambda e , \lambda e]\subseteq [- T^+\lambda e , T^+\lambda e]\). By [2, Theorem 4.9], \([- T^+\lambda e , T^+\lambda e]\) is weakly compact, and so there is a subsequence \(( T^{+}x_{n_{j}})\) of \((T^{+}x_{n})\) which is weakly convergent to some point \(z \in F\). Since \((T^{+}x_{n})\) is an increasing sequence, \((T^{+}x_{n})\) is norm convergent to z, so \(T^{+}\in W_b(E,F)\). A similar argument reveals that \(T^{-}\in W_b(E,F)\). We thus conclude that \(T=T^{+}-T^{-} \in W_b(E,F)\). \(\square \)
References
Abramovich, Y., Sirotkin, G.: On order convergence of nets. Positivity 9, 287–292 (2005)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)
Alpay, S., Altin, B., Tonyali, C.: On property (b) of vector lattices. Positivity 7, 135–139 (2003)
Alpay, S., Altin, B., Tonyali, C.: A note on Riesz spaces with property-b. Czechoslovak Math. J. 56, 765–772 (2006)
Alpay, S., Altin, B.: A note on \(b\)-weakly compact operators. Positivity 11, 575–582 (2007)
Alpay, S., Altin, B.: On Riesz spaces with \(b\)-property and \(b\)-weakly compact operators. Vladikavkaz. Mat. Zh. 11, 19–26 (2009)
Aqzzouz, B., Elbour, A.: On the weak compactness of \(b\)-weakly compact operators. Positivity 14, 75–81 (2010)
Aqzzouz, B., Moussa, M., Hmichane, J.: Some Characterizations of \(b\)-weakly compact operators. Math. Rep. 62, 315–324 (2010)
Gao, N., Xanthos, F.: Unbounded order convergence and application to martingales without probability. J. Math. Anal. Appl. 415, 931–947 (2014)
Meyer-Nieberg, P.: Banach Lattices, Universitex. Springer, Berlin (1991)
Acknowledgements
We would like to thank the anonymous referee for his/her very careful reading of the manuscript and bring to our attention the reference [6] which significantly improved the results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hamid Reza Ebrahimi Vishki.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mousavi Amiri, M., Haghnejad Azar, K. Some Notes on b-Weakly Compact Operators. Bull. Iran. Math. Soc. 46, 1533–1538 (2020). https://doi.org/10.1007/s41980-019-00340-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-019-00340-1