1 Introduction
We recall from [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Definition 3.1] that an operator
$ T: E \rightarrow Y $
from a Banach lattice
$ E $
to a Banach space
$ Y $
is called a U-Dunford–Pettis operator if
$ \| Tx_n \| \rightarrow 0 $
for every order bounded weakly null sequence
$ (x_n) $
in
$ E $
. Alternatively,
$ T $
maps order bounded weakly relatively compact subsets of
$ E $
to relatively compact subsets of
$ Y $
(see [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Theorem 3.2]).
Furthermore, as defined in [Reference Aliprantis and Burkinshaw2, Section 2, p. 231], an operator
$ T $
from the Banach space
$ Z $
to the Banach space
$ X $
is called weak Dunford–Pettis if for every weakly null sequence
$ (x^{\prime }_n) $
in
$ X' $
and every weakly null sequence
$ (z_n) $
in
$ Z $
, it holds that
$ \lim _{n \to \infty } x^{\prime }_n(Tz_n) = 0 $
. Alternatively,
$ T $
maps a weakly compact subset of
$ Z $
onto a Dunford–Pettis set in
$ X $
.
From [Reference Aqzzouz and Bourass5, Section 1, p. 186] we define an operator
$ T: E \to X $
to be order Dunford–Pettis if it maps order bounded subsets onto Dunford–Pettis sets in
$ X $
. In this article, we introduce a new class of operators from a Banach lattice to a Banach space, nominated “weak U-Dunford–Pettis operators” (abbreviated as weak U-DP), which generalizes the classes of U-Dunford–Pettis operators, weakly Dunford–Pettis operators (abbreviated as weak DP), and order Dunford–Pettis operators (abbreviated as order DP). This article is organized as follows: First, we defined the weak U-DP operator in Definition 3.1. Then, we used some examples to show the strict inclusions between the class of weak U-DP operators and other classes of operators. In Theorem 3.4, we present a characterization of the new class. In Corollaries 3.5 and 3.6, we give a sufficient and necessary condition on Banach lattices such that for every operator (or order bounded operator) is weak U-Dunford–Pettis. In Theorem 3.10, we establish sufficient conditions under which certain lattice properties in Banach lattices coincide. Finally, Propositions 3.14, 3.16, 3.18, and 3.19 and Theorem3.20 provide necessary or sufficient conditions for a weak U-DP operator to be classified as U-DP, weak DP, and order DP, respectively.
2 Preliminaries
A norm bounded subset
$ A $
of a Banach space
$ X $
is called a Dunford–Pettis set if, for every weakly convergent sequence
$ (x^{\prime }_n) $
in the dual space
$ X' $
, we have
$\sup _{x \in A} |x^{\prime }_n(x)| \to 0 $
(see [Reference Andrews4, Theorem 1]). A Banach space X has the Dunford–Pettis property (or simply DPP), if every weakly compact operator T from X into another Banach space Y is Dunford–Pettis operator ([Reference Andrews4, Theorem 1]), if and only if for every weakly null sequence
$(y^{\prime }_{n})$
of
$X'$
and weakly null sequence
$(x_{n})$
of X we have
$y^{\prime }_{n}(x_{n})\overset {w}{\to }0$
[Reference Aqzzouz and Bourass5, Propositions 2.2 and 2.3]. A Banach lattice E has the U-Dunford–Pettis property (or simply UDPP), if each weakly compact operator
$T : E \to Y$
is U-Dunford–Pettis for each Banach space Y. To illustrate,
$c_0$
has
$UDPP$
. In fact, for each weakly compact operator
$T:c_0\to Y$
, by applying [Reference Meyer-Nieberg11, Proposition 3.1] the operator
$T=T\circ Id_{c_0}$
is U-Dunford–Pettis because
$Id_{c_0}$
is U-Dunford–Pettis. This is due to the fact that
$c_0$
is discrete and its norm is order continuous, as outlined in [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Proposition 3.1]. We note that
$L^2[0,1]$
does not have
$UDPP$
. This is due to the fact that
$Id_{L^2[0,1]}$
is a weakly compact operator. Furthermore, as
$L^2[0,1]$
is not discrete, as per [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Proposition 3.1],
$Id_{L^2[0,1]}$
is not U-Dunford–Pettis.
A Banach lattice E is said to have the order UDPP (or simply o-UDPP) if every order weakly compact operator
$T : E \to Y$
is U-Dunford–Pettis for every Banach space Y (see [Reference Meyer-Nieberg11, Definition 3.2]). Note that if a Banach lattice is o-UDPP, it is necessarily UDPP, (
$L^2[0.1]$
does not have o-UDPP, and
$l^\infty $
has o-UDPP). A Banach lattice E is said to have the AM-compactness property, if every weakly compact operator defined on E into a Banach space X is AM-compact [Reference Aqzzouz and Bourass6, Section 3, p. 169]. The lattice operations of a Banach lattice E are said to be weakly sequentially continuous, if every weakly null sequence
$(x_\alpha )$
in
$E,$
we have
$\left | x_\alpha \right | \overset {w}{\to }0$
,
$ x^{+}_\alpha \overset {w}{\to }0,$
and
$ x^{-}_\alpha \overset {w}{\to }0$
. An operator
$T:X\rightarrow Y $
between two Banach spaces called a Dunford–Pettis, if
${\left \| Tx_{n}\right \| \to 0}$
for all weakly null sequence
$(x_{n})$
in X (see [Reference Aliprantis and Burkinshaw3, Section 5.4, p. 340]). An operator
$R : G \to X$
from a normed vector lattice G into a Banach space X is order weakly compact if, and only if,
$\left \| S (z_n)\right \|\to 0$
as for every order bounded weak null sequence
$(z_n)_n$
in the positive cone
$G^+$
(see [Reference Meyer-Nieberg11, Corollary 3.4.9]).
We need to establish some notations and definitions. A Banach lattice is a Banach space
$(E,\left \| \right \| )$
such that E is a vector lattice and its norm satisfies the following property: for each
$x, y \in E$
such that
$|x|\leq |y|$
, we have
$\left \|x \right \| \leq \left \| y\right \| $
. Note that if E is a Banach lattice, then its topological dual
$E'$
, endowed with the dual norm and the dual order, is also a Banach lattice. A norm
$\left \| .\right \| $
of a Banach lattice G is order continuous if, for any generalized sequence
$(z_{\beta })$
such that
$z_{\beta }\downarrow 0$
in E,
$(z_{\beta })$
converges to 0 for the norm
$\left \|. \right \| $
, where the notation
$z_{\beta } \downarrow 0 $
means that
$(z_{\beta })$
is decreasing, its infimum exists, and
$inf(z_{\beta }) = 0$
. An operator
$T : E\to F $
between two Banach lattices is a bounded linear mapping.
-
• It is positive if $0\leq T (x) $
in F whenever
$0 \leq x$
in E. -
• It is order bounded if $T(A)$
is order bounded in F for every order bounded subset A in E. -
• It is regular if there are two positive operators $T_{1},T_{2}$
such that
$T=T_{1}-T_{2}$
.
We note by:
-
• $DP_{u}(E,Y)$
: The collection of all U-Dunford–Pettis operators from Banach lattice E to Banach space Y. -
• $DP_{o}(E,Y)$
: The collection of all order Dunford–Pettis operators from Banach lattice E to the Banach space Y. -
• $wDP(E,Y)$
: The collection of all weak Dunford–Pettis operators from Banach lattice E to the Banach space Y.
3 Main results
3.1 The class of weak U-Dunford–Pettis
Definition 3.1 An operator
$T:E\rightarrow Y$
acting between the Banach lattice E and the Banach space Y is said to be weak U-Dunford–Pettis (abb. weak U-DP) if
${\left \langle Tx_{n}, y^{\prime }_{n}\right \rangle \rightarrow 0}$
for every order bounded weakly null sequence
$(x_{n})$
in E and for every weakly null sequence
$(y^{\prime }_{n})$
in
$Y'$
.
We denote the collection of all weak U-DP operators from Banach lattice E to Banach space Y by
$wDP_{u}(E,Y)$
. We note that
$ DP_u(E,Y) \subset wDP_{u}(E,Y) $
, but the reverse inclusion is not generally true. In fact, let
$ (r_n) $
denote the sequence of Rademacher functions on
$ [0, 1] $
, and let
$ S: L^1[0, 1] \to l^{\infty } $
be the positive operator defined by
Given that
$ r_n \stackrel {w}{\to } 0 $
,
$ (r_n) \subset [-1, 1] $
, and
$ \| S(r_n) \| \geq \int _{0}^{1} r_n(t) r^+_n(t) dt = \frac {1}{2} $
, we observe that
$ S $
is not a U-Dunford–Pettis operator. However, because
$ L^1[0, 1] $
possesses the Dunford–Pettis property, it follows that
$ S $
is a weak Dunford–Pettis operator. Therefore,
$ S $
is a weak U-DP operator.
Additionally, we have that
$ DP_o(E,Y) $
is strictly included in
$ wDP_{u}(E,Y) $
. Indeed, the natural embedding
$ J: L^{\infty }[0,1] \to L^{2}[0,1] $
is a positive U-Dunford–Pettis operator but is not AM-compact (see [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Remark 3.4]). Therefore, it is weak U-DP (since
$ DP_u(E,Y) \subset wDP_u(E,Y) $
). Moreover,
$ J $
fails to be an order Dunford–Pettis operator. If the operator
$ J = Id_{L^{2}[0,1]} \circ J $
were AM-compact (see [Reference Aqzzouz and Bourass5, Theorem 2.7]), it would lead to a contradiction.
We also have that
$ wDP(E,Y) \subset wDP_{u}(E,Y) $
strictly. In fact, the operator
$ Id_{l^{2}} $
is weak U-DP. This is due to the fact that
$ l^{2} $
has the AM-compactness property, which implies that
$ Id_{l^{2}} $
is order Dunford–Pettis (see [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.4]). Although
$ Id_{l^{2}} $
is weak U-DP, it is not weak Dunford–Pettis. If that’s not the case, then
$ Id_{l^2} $
is a Dunford–Pettis operator. This is because
$ Id_{l^2} $
is weakly compact (since
$ l^2 $
is reflexive) and satisfies
$ Id_{l^2} = Id_{l^2} \circ Id_{l^2} $
(as shown in [Reference Aliprantis and Burkinshaw3, Theorem 5.99]). This means that the unit ball of
$ l^2 $
would be relatively compact, which is a contradiction since
$ l^2 $
is infinite-dimensional.
Now, we will mention some algebraic properties related to this class of operators.
Proposition 3.1 Let E and F be two Banach lattices and X and Y be two Banach spaces. We have the following assertions:
-
(1) If $T:E\longrightarrow X$
is a weak U-DP operator, then
$S\circ T$
is weak U-DP operator for every operator
$S:X\longrightarrow Y$
. -
(2) If $T:F \longrightarrow Y$
is a weak U-DP operator and
$S:E\longrightarrow F$
is an order bounded operator, then
$T\circ S$
is weak U-DP operator.
Proof The proof is straightforward.
We note that
$ L^2[0,1] $
does not have the AM-compactness property, and its lattice operations are not weakly sequentially continuous. The identity operator
$ Id_{L^2[0,1]} $
, although positive, is not a weak U-DP operator. However, since
$ L^2[0,1] $
is reflexive, theorem 3.4 states that
$ Id_{L^2[0,1]} = Id_{L^2[0,1]} \circ Id_{L^2[0,1]} $
is U-Dunford–Pettis. This leads to the conclusion that
$ L^2[0,1] $
must be discrete (as shown in [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.1]), which is a contradiction. We will now outline the sufficient conditions on lattices that ensure an operator is weak U-DP.
Proposition 3.2 Let E and F be two Banach lattices, and let Y be a Banach space.
-
(1) If the lattice operations of E are weakly sequentially continuous, then every operator $T:E\to Y$
is weak U-DP. -
(2) If the lattice operations of F are weakly sequentially continuous, then every order bounded operator $T:E\to F$
is weak U-DP. That is,
$L_{b}(E,F)\subset wDP_{u}(E,F)$
. -
(3) If F has the AM-compactness property, then every regular operator $T:E\to F$
is weak U-DP. Alternatively,
$L_{r}(E,F)\subset wDP_{u}(E,F)$
.
Proof
-
(1) Let $(x_{n})$
be an order bounded weak null sequence of E such that E has weakly sequentially continuous lattice operations, and T be an operator from E to another Banach space Y. Then,
$x_{n}\overset {w}{\longrightarrow }0 \implies x_{n}\overset {\left | w\right | }{\longrightarrow }0,$
which implies that
$\left \lbrace 0,x_{1},x_{2},\ldots \right \rbrace $
is order
$\left | \sigma \right |- $
weakly compact subset of E. Hence, by [Reference Aqzzouz and Bourass5, Theorem 2.6],
$A=\left \lbrace 0,x_{1},x_{2},\ldots \right \rbrace $
is a Dunford–Pettis set of E. So
$T( A)=\left \lbrace 0,T(x_{1}),T(x_{2}),\ldots \right \rbrace $
is also a Dunford–Pettis set of Y. -
(2) Let $(x_{n})$
be an order bounded weakly null sequence of E and
$T:E\to F$
be an order bounded operator. Then,
$(T(x_n))$
is an order bounded weakly null sequence in F. Since F has weakly sequentially continuous lattice operations, then
$B=\left \lbrace 0,T(x_{1}),T(x_{2}),\ldots \right \rbrace $
is an order
$\left | \sigma \right |- $
weakly compact subset of F. Hence, by [Reference Aqzzouz and Bourass5, Theorem 2.6], B is a Dunford–Pettis set of F. -
(3) Let $T:E\to F$
be a regular operator. Then, there exist two positive operators
$T_{1},T_{2}:E\to F$
such that
$T=T_{1}-T_{2}$
.Now, let $(x_{n})$
be an order bounded weakly null sequence in E. Then, there exists
$x\in E^{+}$
such that
$(x_{n})\subset [-x,x]$
, and let
$(f_{n})$
be a weakly null sequence in
$F'$
. We have $$ \begin{align*}|f_{n}(Tx_{n})|\leq |f_{n}|(|Tx_{n}|) \leq |f_{n}|(\left| T_{1}x_n\right| )+|f_{n}|(\left| T_{2}x_n\right| ).\end{align*} $$
Since F satisfies the AM-compactness property, by [Reference Aqzzouz and Bourass5, Theorem 3.7], for all $y\in F$
,
$|f_{n}|(y)\to 0$
. This implies that $$ \begin{align*}|f_{n}|(|T_{i}x_{n}|)\leq |f_{n}|\left( |T_{i}|(|x_{n}|)\right) \leq |f_{n}|(T_{i}x)\to 0,\quad i=1,2.\end{align*} $$
So $|f_{n}(Tx_{n})|\to 0$
, showing that T is weak U-DP.
Remark 3.3 In general, if a Banach lattice E has the AM-compactness property, its lattice operations are not necessarily weakly sequentially continuous. For example, the Banach lattice
$L^{1}[0,1]$
has the AM-compactness property, but its lattice operations are not necessarily weakly sequentially continuous. Conversely, the weakly sequentially continuity of the lattice operations of E does not imply that E admits the AM-compactness property. Indeed, the lattice operations of
$l^{\infty }$
are weakly sequentially continuous, but
$l^{\infty }$
does not have the AM-compactness property (see [Reference Aqzzouz, Bourass and Moussa7, Remark 2.2(4)]).
The next result characterizes the weak U-DP operator and it is an analog of [Reference Aliprantis and Burkinshaw3, Theorem 5.99].
Theorem 3.4 For an operator
$T : E \to X$
from a Banach lattice E to a Banach space X, the following statements are equivalent:
-
(1) T is a weak U-DP operator.
-
(2) T carries order bounded relatively weakly compact subsets of E to Dunford–Pettis subsets of X.
-
(3) If S is a weakly compact operator from X to an arbitrary Banach space, then $S\circ T$
is a U-Dunford–Pettis operator.
Proof The proof is similar to that of [Reference Aliprantis and Burkinshaw3, Theorem 5.99].
In the following results, we give a some characterization of the weak U-DP class of operators, using the UDPP that it is defined in [Reference Elbour and Aqzzouz10, Definition 3.2], as well as in Corollaries 3.5 and 3.6.
Corollary 3.5 Let E be a Banach lattice. The following assertions are equivalent:
-
(1) Every operator $S:E\to Y$
is weak U-DP for every Banach space Y. -
(2) $Id_{E}$
is weak U-DP. -
(3) E has UDPP.
Proof
$(1)\implies (2)$
is obvious.
$(2)\implies (1)$
Let
$S:E\to Y$
be an operator, since
$Id_E$
is weak U-DP operator then
$S=S\circ Id_E$
is also a weak U-DP operator (see Proposition 3.1).
$(2)\implies (3)$
Let
$S:E\to Y$
be a weakly compact operator, since
$Id_E$
is a weak U-DP operator then
$S=S\circ Id_E$
is also a U-Dunford–Pettis operator (see Theorem 3.4 (
$1)\implies (3)$
), then E has UDPP.
$(3)\implies (2)$
Let
$S:E\to Y$
be a weakly compact operator and E has UDPP then S is a U-Dunford–Pettis operator, since
$S=S\circ Id_E$
then by (
$3)\implies (1)$
of Theorem 3.4,
$Id_E$
is weak U-DP.
We have also similarly the following corollary.
Corollary 3.6 Let F be a Banach lattice. The following assertions are equivalent:
-
(1) Every order bounded operator $S:E\to F$
is weak U-DP for every Banach lattice E. -
(2) $Id_{F}$
is weak U-DP. -
(3) F has UDPP.
By using Corollary 3.5 and Definition 3.1, we can give an explicit version of the UDPP.
Corollary 3.7 Let E be a Banach lattice. The following assertions are equivalent:
-
(1) E has UDPP.
-
(2) For every order bounded weakly null sequence $(x_{n})$
in E, and every weakly null sequence
$(f_{n})$
in
$E'$
, we have
$f_{n}(x_{n})\to 0$
. -
(3) For every order bounded weakly null sequence $(x_{n}\!)$
in E, the set
$\left \lbrace x_{1},x_{2},\ldots ,x_{n},\ldots \right \rbrace $
is a Dunford–Pettis set.
As consequence of Corollary 3.5 and Proposition 3.2, we have the following result.
Corollary 3.8 Let E be a Banach lattice. If the lattice operations of E are weakly sequentially continuous or E has the AM-compactness property, then E has the UDPP.
Proof
-
(1) If the lattice operations of E are weakly sequentially continuous, then the positive operator $Id_{E}$
is weakly U-DP (see Proposition 3.2) and by Corollary 3.5, E has UDPP. -
(2) If E has the AM-compactness property, then $Id_{E}$
is order Dunford–Pettis (see [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.1]). Then, it is weak U-DP, so E has UDPP.
Remark 3.9 We note that if
$ E $
is a Banach lattice with an order unit
$ e $
, then
${B_E = [-e, e] }$
. In this case, every order weakly compact operator from
$ E $
to any Banach space
$ Y $
is also weakly compact. Which implies that
$ E $
has the UDPP, if and only if
$ E $
has the o-UDPP. According to [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Proposition 3.2],
$ E $
has o-UDPP if and only if the lattice operations of
$ E $
are weakly sequentially continuous, since every weakly null sequence in
$ E $
is also an order bounded weakly null sequence.
It is obvious that if E has o-UDPP, then it has UDPP. But the converse is not generally true. In fact, if we take
$E=L^1[0,1]$
, then
$E'=L^{\infty }[0,1]$
, since the lattice operations of
$E'$
are weakly sequentially continuous, then E has AM-compactness property according to [Reference Aliprantis and Burkinshaw2, Theorem 3.3], which implies that
$Id_E$
is order DP (see [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.4]), so it is weak U-DP, thus E has UDPP (see Corollary 3.5). In addition, E does not have o-UDPP. In fact, the operator
$ S: L^1[0, 1] \to l^{\infty } $
defined in Section 3.1 is an order weakly compact operator, but not a U-DP operator. Note that the norm of
$(L^1[0,1])'=L^{\infty }[0,1]$
is not order continuous. Now, we will give some other important characterizations of the UDPP on a Banach lattice.
Theorem 3.10 Let E be a Banach lattice such that the norm of E and the norm of
$E'$
are order continuous, then the following statements are equivalent:
-
(1) E has the UDPP.
-
(2) E has the AM-compactness property .
-
(3) E has the o-UDPP.
-
(4) E is discrete.
-
(5) The lattice operations of E are weakly sequentially continuous.
Proof
-
• (1) $\implies $
(2) If E has UDPP, then
$Id_E$
is weak U-DP, since the norm of E is order continuous(see Proposition 3.20), then
$Id_E$
is order DP, implying that E has the AM-compactness property (see [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.4]). -
• (2) $\implies $
(1) (see Corollary 3.8). -
• $(1)\implies (3)$
Let us demonstrate this by contradiction: let us assume that E has the UDPP but not the o-UDPP. According to [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Proposition 3.2], there exists a weakly null and order bounded sequence
$(x_n)$
in E such that
$ |x_n| \stackrel {w}{\nrightarrow } 0 $
. This implies the existence of a positive functional
$f \in E^{\prime }{+} $
such that
$ f(|x_n|) \nrightarrow 0 $
. Consequently, for any
$\epsilon> 0$
, there exists a subsequence
$(y_n)$
of
$( (x_n) )$
such that
$f(|y_n|)> \epsilon $
for all
$n \in \mathbb {N}^{\star } $
.We have
$f(|y_n|) = \sup \left \lbrace |g(y_n)| :g\in E', |g| \leq f \right \rbrace $
,which implies that there exist functional $g_m \in E'$
with
$|g_m| \leq f $
such that: $$ \begin{align*}f(|y_n|) \leq |g_n(y_n)| + \frac{1}{n} \quad \text{for all } n \in \mathbb{N}^{\star}.\end{align*} $$
Since the norm of $ E'$
is order continuous and (
$|g_n| \leq f $
), the sequence
$( (g_n) )$
must have a weakly convergent subsequence
$ (g_{\phi (n)})$
, converging to some
$ g $
(i.e.,
$( g_{\phi (n)} \stackrel {w}{\to } g ))$
. Thus, we also have $$ \begin{align*}\epsilon < f(|y_{\phi(n)}|) \leq |g_{\phi(n)}(y_{\phi(n)})| + \frac{1}{\phi(n)} \leq |(g_{\phi(n)} - g)(y_{\phi(n)})| + |g(y_{\phi(n)})| + \frac{1}{\phi(n)}\to 0.\end{align*} $$
(This conclusion follows because E has the UDPP, and since $(g_{\phi (n)} - g) \overset {w}{\to } 0$
in
$(E')$
and
$y_{\phi (n)} \overset {w}{\to } 0$
( which means
$(|g(y_{\phi (n)})| \to 0))$
, in addition,
$(y_{\phi (n)})$
is order bounded in E, which implies
$(g_{\phi (n)} - g)(y_{\phi (n)}) \to 0;$
see Corollary 3.7).However, this is a contradiction, as we initially assumed that the sequence $(x_1, x_2, \ldots )$
forms a Dunford–Pettis set (see [Reference Aqzzouz and Bourass5, Proposition 2.2]). -
• (3) $\implies $
(1) (obvious). -
• (3) $\implies $
(4): Given that the norm of E is order continuous, it follows that
$Id_E$
is order weakly compact. As E has o-UDPP, we can conclude that
$Id_E$
is U-DP, which in fact implies that E is discrete (see [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.1]). -
• (4) $\implies $
(5) If E is discrete, then
$E'$
is discrete (see [Reference Aliprantis and Burkinshaw2, Corollary 2.3]), which implies that the lattice operations of E are weakly sequentially continuous (see [Reference Aqzzouz and Elbour8, Corollary 2.2]). -
• (5) $\implies $
(3): Let
$ T $
be an order weakly compact operator from
$ E $
to an arbitrary Banach space
$ X $
. Since the lattice operations of E are weakly sequentially continuous,
$ T $
is a U-DP operator. In fact, for any order bounded weakly null sequence
$ (x_n) $
, we have $$\begin{align*}\| Tx_n \| = \| Tx_n^+ - Tx_n^- \| \leq \| Tx_n^+ \| + \| Tx_n^- \| \to 0. \end{align*}$$
This follows from [Reference Meyer-Nieberg11, Corollary 3.4.9], which shows that $ T $
is U-DP, and thus
$ E $
has o-UDPP.
Remark 3.11 The order continuity condition of the norm of E is essential. In fact, we assume that
$E=L^{\infty }[0,1]$
. Note that
$E'=(L^{\infty }[0,1])'$
is order continuous, but E is not. In this case, note that E has UDPP, o-UDPP and the lattice operations of E are weakly sequentially continuous. But E is not discrete and does not have AM-compactness property.
In the following result, we study the domination problem for the class of weak U-DP operators.
Theorem 3.12 If a positive operator
$S:E\to F$
is dominated by a weak U-DP operator
$T:E\to F$
, then S is a weak U-DP operator.
Proof : Let
$ S, T : E \to F $
be two positive operators between Banach lattices such that
$ 0 \leq S \leq T $
, with
$ T $
being a weak U-DP operator. Let
$ A $
be an order bounded relatively weakly compact subset of
$ E $
.
By Theorem 3.4,
$ T(A) $
is a Dunford–Pettis set. Therefore, for every weakly null sequence
$ (y^{\prime }_n) $
in
$ F' $
, it converges uniformly to zero on the set
$ T(A) $
(as established in [Reference Elbour and Aqzzouz10, Theorem 1]).
Additionally, since
$ S(A) \subset T(A) $
(because
$ 0 \leq S \leq T $
implies
$ S[-x, x] \subset T[-x, x] $
for all
$ x \in E^+ $
), we have
This implies that
$ (y^{\prime }_n) $
converges uniformly to zero on the set
$ S(A) $
. Therefore, we conclude that
$ S $
is a weak U-DP operator.
3.2 The relationship between the class of weak U-Dunford–Pettis and other classes
We note that:
-
• $Id_{L^{\infty }[0,1]}$
is weak U-DP (because the lattice operations of
$L^{\infty }[0,1]$
are weakly sequentially continuous). However,
$Id_{L^{\infty }[0,1]}$
is not an order weakly compact operator, and therefore it is not a weakly compact operator. -
• Furthermore, $Id_{L^{2}[0,1]}$
is a weakly compact (respectively, order-weakly compact) operator, but it is not weak U-DP.
Similar to the DPP, UDPP, and o-UDPP properties, we can define new properties on Banach lattices.
Definition 3.2
-
(1) A Banach lattice $ G $
is said to have the weak UDPP (wUDPP) if, for every weakly compact operator
$ S: G \to X $
, the operator
$ S $
is weak U-DP for any Banach space
$ X $
. -
(2) A Banach lattice $ G $
is said to have the order weak UDPP (owUDPP) if, for every order weakly compact operator
$ S: G \to X $
, the operator
$ S $
is weak U-DP for any Banach space
$ X $
. -
(3) A Banach lattice $ G $
is said to have the reciprocal weak UDPP (RwUDPP) if, for every weak U-DP operator
$ S: G \to X $
, the operator
$ S $
is weakly compact for any Banach space
$ X $
. -
(4) A Banach lattice $ G $
is said to have the reciprocal order weak UDPP (RowUDPP) if, for every weak U-DP operator
$ S: G \to X $
, the operator
$ S $
is order weakly compact for any Banach space
$ X $
.
Remark 3.13 It is evident that if a Banach lattice E has o-UDPP, then it must also have owUDPP. For example, we may consider the Banach lattices
$l^p$
, where
$1\leq p <\infty $
. These have owUDPP, however,
$L^2[0,1]$
does not possess this property. This is due to the fact that
$Id_{L^{2}[0,1]}$
is order weakly compact and is not weak U-DP.
3.2.1 Comparison of the weak U-DP operator with the U-DP and the weak DP operators
Now, we give necessary and sufficient conditions for a weak U-DP operator to be U-Dunford–Pettis.
Proposition 3.14 If any positive weak U-DP operator T from E to F is U-Dunford–Pettis, then one of the following assertions is valid:
-
(1) E has o-UDPP.
-
(2) The norm of F is order continuous.
Proof If we assume that neither E has o-UDPP nor the norm of F is order continuous, then [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Theorem 3.1] proves that there exist
$S,T:E\to F$
such that
${0\leq S\leq T}$
, T is U-Dunford–Pettis and S is not U-Dunford–Pettis.
Since T is a U-Dunford–Pettis operator, it is also weak U-DP. By applying Theorem 3.12, we can also conclude that S is weak U-DP. However, our hypothesis implies that S is U-Dunford–Pettis, which is a contradiction.
Remark 3.15
-
• Condition 2) of Proposition 3.14 is not sufficient. Indeed, for $E=F=L^1[0,1]$
, we have that
$Id_{L^1[0,1]}$
is weak U-DP, since it admits UDPP. However, it is not U-Dunford–Pettis because
$L^1[0,1]$
is not discrete (see [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.1]), although the norm of
$L^1[0,1]$
is order continuous. -
• Similarly, condition 1) of Proposition 3.14 is not sufficient. For example, if we take $E = F = c$
, we find that the identity operator
$Id_c$
is weak U-DP. This is because the space c has the UDPP due to the continuity of its lattice operations. Consequently, it also satisfies the o-UDPP, which implies it has the UDPP. However, c does not possess the RowUDP property, since
$Id_c$
is not order weakly compact, as its norm is not order continuous. Additionally,
$Id_c$
is not U-Dunford–Pettis either, because its norm fails to be order continuous (see [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.1]).
Proposition 3.16 Let E and F be two Banach lattices.
-
(1) If $ E $
has the o-UDPP and the RowUDPP, then every weak U-DP operator
$ T $
from
$ E $
to every Banach space
$ X $
is U-DP. -
(2) If $ F $
is discrete and its norm is order continuous, then for every order bounded weak U-DP operator
$ T: G \to F $
, it follows that
$ T $
is U-Dunford–Pettis for every Banach lattice
$ G $
.
Proof
-
(1) Let $ T: E \to X $
be a weak U-DP operator. Given that
$ E $
satisfies the de Row UDPP condition, it follows that
$ T $
is an order weakly compact. Moreover, since
$ E $
also satisfies the o-UDPP condition, we conclude that
$ T $
is a U-Dunford–Pettis operator. -
(2) Let $ T: G \to F $
be an order bounded weak U-DP operator. If
$ F $
is a discrete space with an order continuous norm, then the identity operator
$ Id_F $
is U-Dunford–Pettis (as shown in [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.2]). Since
$ T = Id_F \circ T $
, we can apply [Reference Aboutafail, El Fahri, H’Michane and Zraoula1, Proposition 3.1] to conclude that
$ T $
is also U-Dunford–Pettis.
Other sufficient conditions are given in the following.
Proposition 3.17 Let
$E $
be a Banach lattice and Y be a Banach space, if Y is reflexive, then every weak U-DP operator
$T:E\to Y$
is U-Dunford–Pettis.
Proof Obvious (
$T=Id_{Y}\circ T$
and
$Id_{Y}$
is weakly compact, by Proposition 3.1, T is U-DP).
We now provide sufficient conditions for weak U-DP operators to be weak Dunford–Pettis operators in the following result.
Proposition 3.18 Let
$ S: E \to Z $
be a weak U-DP operator, where
$ E $
is a Banach lattice that admits an order unit, and
$ Z $
is a Banach space, then
$ S $
is a weak Dunford–Pettis operator.
Proof Assume that a Banach lattice
$ E $
has an order unit
$ e $
, and let
$ S $
be a weak U-DP operator from
$ E $
to a Banach space
$ Z $
. We need to show that
$ S $
is a weak Dunford–Pettis operator, which is equivalent to proving that for any relatively weakly compact subset
$ A $
of
$ E $
, the image
$ S(A) $
is a Dunford–Pettis set in
$ Z $
. Since
$ A $
is norm-bounded in
$ E $
, it is also order-bounded. This is because there exists
$ r> 0 $
such that
$ A \subset rB_E = r[-e, e] $
. Given that
$ S $
is a weak U-DP operator, it follows that
$ S(A) $
is a Dunford–Pettis set in
$ Z $
(see Theorem 3.4).
3.2.2 Comparison of the weak U-DP operator with the order DP operator
We give sufficient conditions for a weak U-DP operator from Banach lattice to Banach space to be an order DP operator.
Proposition 3.19 Let E be a Banach lattice and Y be a Banach space. Then, every weak U-DP operator
$T: E \to Y$
is an order Dunford–Pettis operator. Whenever one of the following assertions is valid:
-
(1) The norm of E is order continuous.
-
(2) E has the AM-compactness property.
-
(3) Y has the DPP and is reflexive.
Proof Let
$T: E \to Y$
be a weak U-DP operator.
-
(1) If the norm of E is order continuous, then for all $x\in E^+$
, the
$[-x,x]$
is an order bounded relatively weakly compact subset of E by the hypothesis that
$T([-x,x])$
is a Dunford–Pettis set of F, so T is order Dunford–Pettis. -
(2) If E has AM-compactness property then $Id_{E}$
is order Dunford–Pettis operator (see [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.4]) and since that
$T=T\circ Id_{E}$
then T is order Dunford–Pettis (see [Reference Zraoula, Ouba, H’michane and El Wahbi12, Proposition 3.1]). -
(3) Let $u\in E^{+}$
. Then,
$T[-u,u]$
is a norm-bounded set in Y, since Y is reflexive. Therefore,
$T[-u,u]$
is relatively weakly compact (see [Reference Aliprantis and Burkinshaw3, Theorem 3.32]) and by the hypothesis that Y has DPP, we can conclude that
$T[-u,u]$
is a Dunford–Pettis set (see [Reference Aqzzouz and Bourass5, Proposition 2.3]). Thus, T is an order Dunford–Pettis operator.
In the following, we characterize weak U-DP operators between two Banach lattices as order Dunford–Pettis operators under equivalent conditions.
Theorem 3.20 Let E and F be two Banach lattices such that E and F are
$\sigma $
-Dedekind complete and F has the DPP. The following assertions are equivalent:
-
(1) Every weak U-DP operator $T:E\to F$
is order Dunford–Pettis operator. -
(2) Every positive weak U-DP operator $T:E\to F$
is order Dunford–Pettis operator. -
(3) One of the assertions is valid:
-
(a) The norm of E is order continuous.
-
(b) The norm of F is order continuous.
-
Proof It is obvious that (3) implies (1) and (1) implies (2).
To show that (2) implies (3), it is assumed that if the norm of E and the norm of F are not order continuous, then there exists a positive operator
$T:E\to F,$
which is weak U-DP and such that T is not order Dunford–Pettis. Since E and F are
$\sigma $
-Dedekind complete, it follows from [Reference Aliprantis and Burkinshaw3, Theorem 4.56] that there exist two positive projections
$P_{1}:E\to l^{\infty }$
and
$P_{2}:F\to l^{\infty }$
and two canonical injections
$i_{1} :l^{\infty }\to E$
and
$i_{2} :l^{\infty }\to F$
which are all weak U-DP because the lattice operations of
$l^{\infty }$
are sequentially continuous (applying Proposition 3.2). Then, the composition operators
$T=i_{2}\circ P_{1}:E\to F$
are weak U-DP (see Proposition 3.1), but not order Dunford–Pettis (see [Reference Aqzzouz, Bourass and Moussa7, Remark 2.2(4)] and [Reference Bourass, El Kaddouri, H’Michane and Moussa9, Proposition 3.4]). This is clear from the fact that
$Id_{l^{\infty }}=P_{2}\circ T\circ i_{1}$
is not order Dunford–Pettis.
Proposition 3.20 states that if
$E=F,$
we get the following result.
Corollary 3.21 Let E be a
$\sigma $
-Dedekind complete Banach lattice. Then, the following assertions are equivalent:
-
(1) Every weak U-DP operator $T:E\to E$
is an order Dunford–Pettis operator. -
(2) Every positive weak U-DP operator $T:E\to E$
is an order Dunford–Pettis operator. -
(3) The norm of E is order continuous.
3.3 Conclusion
The following diagram provides a summary of the relationships between the different classes of operators that have been the subject of this work:
Long description
A flowchart with nodes in red boxes connected by black and red arrows.
Primary black arrow flow from left to right:
* K (E, Y) points to L sub w c (E, Y) at the top, D P (E, Y) in the middle, and A M (E, Y) at the bottom.
* L sub w c (E, Y) points to L sub o w c (E, Y) at the far right.
* D P (E, Y) points to D P sub u (E, Y) and downward to w D P (E, Y).
* D P sub u (E, Y) points to L sub o w c (E, Y).
* w D P (E, Y) points to w D P sub u (E, Y).
* A M (E, Y) points to D P sub o (E, Y).
* D P sub o (E, Y) points upward to w D P sub u (E, Y).
Red numbered arrows indicating conditional implications:
* (1) A double-headed red arrow between D P sub u (E, Y) and w D P sub u (E, Y).
* (2) A double-headed red arrow between w D P sub u (E, Y) and D P sub o (E, Y).
* (3) A red arrow from w D P sub u (E, Y) back to w D P (E, Y).
* (4) A red arrow from w D P sub u (E, Y) to L sub o w c (E, Y).
* (5) A red arrow from w D P sub u (E, Y) to L sub o w c (E, Y) following a different path.
Legend for red conditions:
* (1) E has o-U D P P and Row U D P P or Y is reflexive.
* (2) E is order continuous or E has A M-compactness or Y is reflexive and has D P P.
* (3) E has order unit.
* (4) E has ow U D P P.
* (5) E has Row U D P P.
