• 1.    Motivation and Introduction

• 2.    Banach–Saks Operators

Let us start with some motivation. Certainly, a Banach lattice is a Banach space with an order structure and of course with some suitable connections between ordered and analysis aspects. So, it is natural to expect some order aspects of Banach lattices and also operators between them. This line has been growing from the early stage of appearance of vector lattices (with the remarkable Riesz–Kantorovich formulae) until now, using different types of convergence with the aid of the order structure beside the analysis aspects, such as uaw-Dunford-Pettis (M -weakly compact) operators, unbounded continuous operators (W M -weakly compact operators) and so on. Recently, there has been some attention to the order structure of some notions that have been introduced and investigated initially in Banach spaces such as the Banach–Saks property and the Grothendieck property. Different types of the mentioned properties have been defined and some interesting results regarding

characterizations of some known spaces in Banach lattice theory such as order continuity or reflexivity have been obtained (see [1–5] for more details). In this paper, we consider operator versions regarding the different types considered for the Banach–Saks property and various concepts related to the Grothendieck property. As our main results, in both sections, we characterize order continuous Banach lattices and reflexive ones in terms of these different classes of bounded operators. All operators in this paper are assumed to be continuous. For a detailed exposition on Banach lattices and operators between them, as well as the related topics, see [5–7].

First, let us recall different types related to the Banach–Saks property for a Banach lattice. Suppose E is a Banach lattice. E is said to have the unbounded Banach–Saks property (UBSP, for short) if for every norm bounded uaw-null sequence (x _n ) C E (i. e., for each u G E ₊ , | x _n | Л u —— 0), there is a subsequence (x n k ) whose Cesaro means is convergent. Moreover, recall that E possesses the disjoint Banach–Saks property (DBSP, for short) if every bounded disjoint sequence in E has a Ces`aro convergent subsequence; E has the disjoint weak Banach– Saks property (DWBSP, in brief) if every disjoint weakly null sequence in E has a Ces´aro convergent subsequence. Furthermore, E possesses the weak Banach–Saks property (WBSP, in brief) if for every weakly null sequence (x n ), it has a subsequence which is Cesaro convergent. Finally observe that E possesses the Banach–Saks property (BSP) if every bounded sequence (x n ) C E has a Cesaro convergent subsequence. For a brief but comprehensive context in this subject, see [8]. Also, for a detailed exposition on these notions, see [5]. Now, let us consider the operator’s versions of these types of the Banach–Saks properties.

Definition 1. Suppose E and F are Banach lattices and T : E — F is a continuous operator. T is said to be

• (i)    the unbounded Banach-Saks operator ( UBSO , for short) if every bounded uaw -null sequence (x n ) C E, (T(x n )) has a Cesaro convergent subsequence; that is, there exists a subsequence (x n k ) of (x n ) such that ( П £ П₌₁ T(x n k )) is norm convergent;

• (ii)    the disjoint Banach-Saks operator ( DBSO , in notation) if every bounded disjoint sequence (x n ) C E, (T(x n )) has a Cesaro convergent subsequence;

• ( iii )    the weak Banach-Saks operator ( WBSO , in notation) if every weakly null sequence (x n ) C E, (T(x n )) has a Cesaro convergent subsequence;

• (iv)    the disjoint weakly Banach-Saks operator ( DWBSO , in notation) if every disjoint weakly null sequence (x n ) C E, (T (x n )) has a Cesaro convergent subsequence;

• (v)    the Banach-Saks operator ( BSO , in brief) if every bounded sequence (x _n ) C E, (T (x n )) has a Cesaro convergent subsequence.

The parts (iii) and (iv) were defined initially in the definition 6.16 from [8]. These notions extend ideas considered in [5, Section 4] as a Banach lattice E has the corresponding Banach–Saks property if and only if so is the identity operator on E . Moreover, note that while E has the corresponding Banach–Saks property and F is any Banach lattice, every continuous operator T : E — F possesses the Banach-Saks operator’s version, as described in Definition 1, respectively. Therefore, we conclude that these concepts are far from being equivalent, in general as described in [5, Section 4]. Before anything, the following easy fact is handy and fruitful.

Lemma 1. Suppose E , F and G are Banach lattices and T : E — F and S : F — G are continuous operators. Then, we have the following facts.

• (i)    If T is UBSO , then so is ST .

• (ii)    If T is DBSO , then so is ST .

• (iii)    If T is WBSO , then so is ST .

• (iv)    If T is DWBSO , then so is ST .

• (v)    If T is BSO , then so is ST .

Now, we extend [5, Lemma 36] to the operator case; the proof has essentially the same idea.

Lemma 2. (i) Suppose E' is order continuous. Then for every Banach lattice F , every WBSO T : E ^ F is UBSO .

• (ii)    For Banach lattices E and F , every UBSO T : E ^ F is also DBSO .

• (iii)    Suppose E is either an AM -space or an atomic order continuous Banach lattice. Then for every Banach lattice F , every UBSO T : E ^ F is WBSO .

For the converse, we have the following.

Theorem 1. Suppose E is a Banach lattice. Then the following are equivalent.

• (i)    E ' is order continuous.

• (ii)    every WBSO T : E ^ 1 1 is UBSO .

^ (i) ^ (ii) is trivial by Lemma 2. For the converse, suppose not. By [7, Proposition 2.3.11], there exists a positive projection P : E ^ l i . P is in fact a WBSO ; suppose (x _n ) is a weakly null sequence in E so that P(x _n ) is weakly null in l i . By the Schur property, it is norm null so that whose Ces`aro means is also norm null, as well. But P is not an UBSO . Suppose (e n ) is the standard basis in l i . It is disjoint in l i so that it is disjoint in E. By [9, Lemma 2], it is uaw -null. Nevertheless, it is easy to see that whose Ces`aro means is not convergent in l i , as claimed. >

It is easy to see that every WBSO T : E ^ F is a DWBSO . Therefore, we have the following.

Corollary 1. Suppose E and F are Banach lattices and every DWBSO T : E ^ F is an UBSO . Then E ^′ is order continuous.

Moreover, for a converse of Lemma 2, part (ii), we have the following.

Proposition 1. Suppose E is an order continuous Banach lattice and F is any Banach lattice. Then every DBSO T : E ^ F is UBSO .

• < 1 Suppose (x n ) is a norm bounded uaw-null sequence in E. By [9, Theorem 4] and [1, Theorem 3.2], there are a subsequence (x n k ) of (x n ) and a disjoint sequence (d k ) such that ||x _n k - d k || ^ 0. By passing to a subsequence, we may assume that lim m ,^ m ^^ T(d i ) ^ 0. Now, the result follows from the following inequality.

  m

  m E ^{T ( x} n i ) i=i

  
  

  _-

  
  

  m

  - Et (d i )

  m

  i=i

  
  

  m

  < mm E»T IHK - d i IH 0. > i=i

  
  

Remark 1. Note that order continuity is essential in Proposition 1 and can not be omitted. Consider the identity operator I on ℓ _∞ ; it is obviously DBSO by [5, Lemma 38], but it fails to be UBSO , the uaw-null sequence (u n ) defined via u _n = (0,..., 0,1,..., 1, 0,...), in which one is appeared n-times (from the n-th position until the 2n-th position), does not have any Ces`aro convergent subsequence; although it is weakly null so that uaw -null by [9, Theorem 7], although it does have the DBSP , certaily.

For a sort of the converse of Proposition 1, we have the following partial results.

Proposition 2. Suppose F is a Banach lattice. If every continuous operator T : l i ^ F is UBSO ( DBSO ) , then F ' is order continuous.

• < 1 First note that since the domain l i has an order continuous norm, UBSO is equivalent by DBSO by Proposition 1 and Lemma 2. Suppose not. By [6, Theorem 4.71], F has a lattice copy of l i . Consider the embedding map from l i into F ; the standard disjoint sequence (e n ) is uaw-null in both l i and E by [9, Lemma 2], however, the Cesaro means is not convergent for any subsequence. >

Proposition 3. Suppose F is a σ -order complete Banach lattice. If every continuous operator T : l ^ 4 F is UBSO , then F is order continuous.

• <    Suppose not. By [6, Theorem 4.56], F has a lattice copy of l ^ . Consider the embedding map from l ^ into F ; now, consider Remark 1 to derive the desired result. >

Now, we consider some ideal properties for these classes of operators. Recall that a continuous operator T : E 4 F , where E and F are Banach lattices, is said to be unbounded continuous provided that it maps every bounded uaw-null sequence (x _n ) C E, to a weakly null sequence; T is uaw-continuous if it maps norm bounded uaw-null sequences into uaw-null ones. For more details, see [5].

Proposition 4. Let E, F and G be Banach lattices. Then the following assertions hold.

• (i)    If T : F 4 G is a DBSO and S : E 4 F is disjoint-preserving, then TS is DBSO .

• (ii)    If T : F 4 G is an UBSO operator and S : E 4 F is uaw -continuous, then TS is UBSO .

• (m)    If T : F 4 G is WBSO and S : E 4 F is an unbounded continuous operator, then TS is also UBSO .

• <    ( i ) Suppose ( x _n ) is a norm bounded disjoint sequence in E . Therefore, by the assumption, (S(x _n )) is also bounded and disjoint in F . Therefore, (TS(x _n )) has a Cesaro convergent subsequence.

• (ii)    Suppose (x _n ) is a norm bounded uaw-null sequence in E. Therefore, S(x n ) —4- 0. Thus, (TS(x _n )) has a Cesaro convergent subsequence.

• (iii)    Suppose (x n ) is a norm bounded uaw-null sequence in E. By the assumption, S(x n ) —4 0 so that (TS (x n )) has a Cesaro convergent subsequence. >

Theorem 2. Suppose E is a Banach lattice, such that both E and E ^′ have order continuous norms and F is any Banach lattice. Then each DWBSO T : E 4 F is UBSO .

• < Suppose T : E 4 F is DWBSO and ( x _n ) is a norm bounded uaw -null sequence in E . By [9, Theorem 4] and [1, Theorem 3.2], there are a subsequence (x n k ) of (x n ) and a disjoint sequence (d k ), such that ||x _n k — d k || 4 0. By [9, Lemma 2], x _n - ^u - ^a 4 ^w 0, so that x _n -4 ^w 0 by [9, Theorem 7]. By passing to a subsequence and by the assumption, we may assume that lim m ,^ m ^ m=i T(d i ) 4 0. Now, the result follows from the following inequality.

  m

  m E ^{T ( x} n i ) ^— i=i

  
  

  mm

  mET⁽d> < mE i^|T ни

  
  

  i =1

  
  

  m

  
  

  i =1

  
  

  x n i ^{— d} i ¹1 ^{4 0} - l>

  
  

sequence (x ‘ _П ) C E ‘ , we have x ’ n -4 0. Note that this equivalent to say that E " is order continuous; for more details, see [10, Corollary 2.9]. It is proved in [4] that a Banach lattice E is refelexive if and only if for every bounded sequence (х П ) C E' , х П ---4 0 implies that х П -4 0. More precisely, we mention the main result from [4] as follows.

Theorem 3. A Banach lattice E is reflexive if and only if every norm bounded uaw ^∗ -null sequence in E ^′ is weakly null.

Now, consider the following definitions regarding operators that enjoy different types considered for the Grothendieck property.

Definition 2. Suppose E and F are Banach lattices and T : E 4 F is a continuous operator. T is said to be:

• (i)    the Grothendieck operator ( GO , for short), if every bounded w * -null sequence (х П ) C F ' , T ' (х П ) 4 0;

• (ii)    the disjoint Grothendieck operator ( DGO , in notation), if every bounded disjoint sequence (х П ) C F ' , T ' (х П ) -4 0;

• (iii)    the unbounded Grothendieck operator ( UGO , in notation), if every norm bounded uaw ^∗ -null sequence (х П ) C F ' , T ' (х П ) -4 0;

The first part of Definition is initially considered in [11]. Furthermore, some different aspects of the Grothendieck property with emphasis on order structure has been studied recently in [4]. Now, we consider some relations between these classes of operators.

Proposition 5. Suppose E and F are Banach lattices, such that F has the GP . Then every continuous operator T : E 4 F is a GO .

• < 1 Suppose (у П ) C F ' is a w * -null sequence in F ' . By the assumption, у П 4 0. This implies that T ' (y'_n ) -4 0 by [6, Theorem 5.22]. >

By using [9, Lemma 2], we conclude that every UGO is DGO . For a sort of the converse, we have the following result; before that, we need the following fact.

Proposition 6. Suppose E is a Banach lattice and F is an order continuous Banach lattice. Then, every GO T : E 4 F is UGO .

• <    (i). Suppose (х ' _П ) is a norm bounded uaw * -null sequence in F ' . By [9, Proposition 5], х ' _П -4 0, so that T ' (х ' _П ) -4 0 in E' by the assumption. >

Corollary 2. Assume that a Banach lattice F is reflexive. Then every continuous operator T : 1 1 4 F is an UGO .

• <    Assume that F is reflexive so that it has GP . In addition, suppose T : l i 4 F is a continuous operator. By Proposition 6, T is an UGO . ⊲

Now, we show some ideal properties for the unbounded Grothendieck operators.

Proposition 7. Assume that E , F and G are Banach lattices. Then we have the following observations.

• (i)    If T : F 4 G is an UGO and S : E 4 F is a continuous operator. Then TS : E 4 G is also UGO .

• (ii)    Suppose T : E 4 F is an UGO and S : F 4 G is interval-preserving, such that S ' is also onto. Then ST is also UGO

• <    (i) Suppose (х П ) is a bounded uaw * -null sequence in G ' . Since (TS )' = S'T ' , we conclude that T ' (х П ) -4 0, so that ST ' (х П ) -4 0.

• (ii) Suppose (х П ) is a bounded uaw * -null sequence in G' . Since S is interval-preserving, by [6, Theorem 2.19], S is a lattice homomorphism. Note that continuity of S implies that it

preserves weak ^∗ -convergence. On the other hand, since S ^′ is onto, we see that for each v ^′ ∈ F ^′ , there exists a u' G G with S ‘ (u') = v ‘ . Therefore,

S ' (x ' n ) Л v ' = S ' (x ' n Л u' ) -A 0.

Thus, (ST ) ' (х П ) = T ' (S ' (х П )) A 0. >