^{1}

^{2}

^{2}

^{1}

^{2}

We prove that an operator is weak Dunford-Pettis if its adjoint is one but the converse is false in general, and we give some necessary and sufficient conditions under which each positive weak Dunford-Pettis operator has an adjoint which is weak Dunford-Pettis.

Let us recall that an operator

The class of weak Dunford-Pettis operators was used by Aliprantis and Burkinshaw [

On the other hand, if

Morever, if

Also, if

As we have already done for Dunford-Pettis operators [

We refer the reader to [

Let us recall that an operator

It follows from [

On the other hand, if

the lattice operations in the topological dual

the lattice operations in

A Banach space (resp., Banach lattice)

We need to recall, from [

Let

the norm of

the topological dual

the lattice operations in

the lattice operations in

There exists a Banach lattice

the norm of

the norm of

the norms of

the norms of

the topological dual

the topological dual

A Banach space

Note that the Schur property implies the Dunford-Pettis property, and hence the weak Dunford-Pettis property, but the weak Dunford-Pettis property does not imply the Schur property. In fact, the Banach space

The following result gives some sufficient conditions for which the topological dual, of a Banach lattice, has the Schur property.

Let

the norm of

the norms of

the topological dual

(1) Let

Now, by Corollary 2.7 of Dodds and Fremlin [

For (2) and (3), it follows from Theorem

(1) There exists a Banach lattice

(2) If the topological dual

Now, we study the duality property of weak Dunford-Pettis operators. Our first result proves that each operator is weak Dunford-Pettis whenever its adjoint is one.

Let

Let

Now, as

Let us recall from [

Now, we give some sufficient conditions for which each positive weak Dunford-Pettis operator has an adjoint which is Dunford-Pettis.

Let

the norm of

the norm of

the norms of

For (1), (2), and (3), let

As

Since

Since the norm of

In this case, each operator

There exist Banach lattices

if the topological dual

if

As a consequence of Theorems

Let

the topological dual

the norm of

the norm of

the norm of

the norms of

the norms of

For (1), (2), (3), (4), and (5), it follows from Theorem

(6) Follows from (3) of Theorem

(7) In this case each operator

For the converse of Theorem

Let

the norm of

Assume by way of contradiction that the norm of

Since the norm of

On the other hand, since

Now, we consider the operator

Then

Let

the norm of

Whenever

Let

each positive weak Dunford-Pettis operator

the norms of

(1)

(2)

In [

Let us recall that an operator

Note that the adjoint of a positive almost Dunford-Pettis operator is not necessarily Dunford-Pettis. In fact, the identity operator of the Banach space

The following result gives some sufficient conditions for which each positive almost Dunford-Pettis operator has an adjoint which is Dunford-Pettis.

Let

the norm of

the norm of

Note that for (1) and (2), the proof is the same as (1) and (2) of Theorem

As

As

Now to prove that

Since the norm of

In this case each operator

Let

If the topological dual

If

If

For the converse of Theorem

Let

the norm of

The proof is the same as that of Theorem

Let

Finally, we note that there exists a positive weak Dunford-Pettis (resp., Dunford-Pettis) operator

Now, we give a characterization on the duality between weak Dunford-Pettis operators and almost Dunford-Pettis operators.

Let

each positive weak Dunford-Pettis (resp., Dunford-Pettis, almost Dunford-Pettis) operator

one of the following assertions is valid:

the norm of

(1)

Since the norm of

On the other hand, since

Now, we consider the composed operator

Since

Then

(2), (a)

On the other hand, since the norm of

(2), (b)