# Yosida-Hewitt type decompositions for order-weakly compact operators.

1 Introduction and preliminariesThe problem of Yosida-Hewitt type decompositions of linear mappings from vector lattices to vector lattices (Banach spaces) has been considered in [E], [S], [[AB.sub.1]], [KM], [BBuY], [BBu]. In particular, Basile, Bukhvalov and Yakubson ([BBuY], [BBu]) have derived Yosida-Hewitt type decompositions for order-weakly compact operators from vector lattices to Banach spaces. Recall here that a linear operator T from a vector lattice E to a Banach space Y is said to be order-weakly compact if the set T([-u, u]) is relatively weakly compact in Y for every u [member of] [E.sup.+] (see [D], [AB.sub.2], [section] 18]). In [n7] we obtained Yosida-Hewitt type decompositions for weakly compact operators from Kothe-Bochner function spaces E(x) to Banach spaces. The purpose of this paper is to derive Yosida-Hewitt type decompositions for order-weakly compact operators acting from more general function spaces E(X) to Banach spaces (see Theorems 3.3, 3.4 and 3.6 below).

We denote by [sigma](L, K) and [tau](L, K) the weak topology and the Mackey topology on L with respect to a dual pair <L,K>. For terminology concerning vector-lattices and function spaces we refer to [[AB.sub.2]], [KA].

Throughout the paper we assume that ([OMEGA], [SIGMA], [mu]) is a complete (T-finite measure space. Let [L.sup.0] denote the space of [mu]-equivalence classes of all [SIGMA]-measurable real valued functions defined on [OMEGA]. Let E be an ideal of [L.sup.0] with supp E = [OMEGA], and let E' stand for the Kothe dual of E. We will assume that supp E' = [OMEGA]. Let [E.sup.~], [E.sup.~.sub.n] and [E.sup.~.sub.s] stand for the order dual, the order continuous dual and the singular dual of E respectively. Then [E.sup.~.sub.n] separates the points of E and it can be identified with E' through the mapping: E' [contains as member] v [??] [[phi].sub.v] [member of] [E.sup.~.sub.n], where [[phi].sub.v]{u) = [[integral].sub.[OMEGA]] u(w)v{w)d[mu] for all u [member of] E.

From now on we assume that (X, [parallel] x [parallel]X), X [not equal to] {0} and (Y, [parallel] x [parallel]y)Y, Y [not equal to] {0} are real Banach spaces and let [X.sup.*] and [Y.sup.*] stand for their Banach duals. Let [S.sub.X] stand for the unit sphere in X. By [L.sup.0](X) we denote the set of [mu]-equivalence classes of all strongly [summation]-measurable functions f : [OMEGA] [right arrow] X. For f [member of] [L.sup.0] (X) let us set [??]([omega]) := [[parallel]/([omega])[parallel].sub.X] for [omega] [member of] [OMEGA]. Let

E(X)= {f [member of] [L.sup.0](X): [??] [member of] E}.

Basic concepts of the theory of vector-valued spaces E(X) can be found in monographs: [CM], [DU], [L]. Recall that the algebraic tensor product E [cross product] X is the subspace of E(x) spanned by the functions of the form u [cross product] x, (u [cross product] x) ([omega]) = u([omega])x, where u [member of] E, x [member of] X. For each u [member of] [E.sup.+] the set [D.sub.u] = {f [member of] E(x) : f [less than or equal to] u} will be called an order interval in E(x) (see [BuL]).

Following [D], [[N.sub.4]], [[N.sub.5]] we are now ready to define two classes of linear operators.

Definition 1.1. A linear operator T : E(X) [right arrow] Y is said to be order-weakly compact (resp. order-bounded) whenever for each u [member of] [E.sup.+] the set T([D.sub.u]) is relatively-weakly compact (resp. norm bounded) in Y.

Clearly each order-weakly compact operator T : E(x) [right arrow] Y is order-bounded.

2 Duality of vector-valued function spaces

In this section we establish terminology and prove some results concerning duality of vector-valued function spaces E(X) (see [BuL], [[N.sub.1]], [[N.sub.2]], [[N.sub.3]], [[N.sub.4]]).

For an order-bounded functional F on E(x) let us put

[absolute value of F](f) := sup{[absolute value of F(h)] : h [member of] E(X), [??] [less than or equal to] [??]} for f [member of] E(X).

Clearly [F(f)] [less than or equal to] [absolute value of F](f) for each f [member of] E(X) and [absolute value of F]([f.sub.1]) [less than or equal to] [absolute value of F](f.sub.2]) whenever [[??].sub.1] [less than or equal to] [[??].sub.2]. One can check that the mapping f [??] [absolute value of F](f) is a seminorm on E(X).

The set

E[(X).sup.~] = {F [member of] E[(X).sup.#] : [absolute value of F](f) < [infinity] for all f [member of] E(X)}

will be called the order dual of E(X) (here E[(X).sup.#] denotes the algebraic dual of E(X)). It is known that a linear operator T : E(X) [right arrow] Y is order bounded if and only if T is ([tau](E(X), E[(X).sup.~]), [[parallel] x [parallel].sub.Y])-continuous (see [[N.sub.4], Theorem 2.3]).

Let F [member of] E[(X).sup.~] and [x.sub.0] [member of] [S.sub.X] be fixed. For u [member of] [E.sup.+] let us set

[[phi].sub.F](u) := [absolute value of F](u [cross product] [x.sub.0]) = sup{[absolute value of F(h)] : h [member of] E(X), [??] [less than or equal to] u).

Note that [[phi].sub.F](u) does not depend on [x.sub.0] [member of] [S.sub.X]. Then [[phi].sub.F] : [E.sup.+] [right arrow] [R.sup.+] is an additive mapping and [[phi].sub.F] has a unique positive extension to a linear mapping from E to R (denoted by [[phi].sub.F] again) and given by

[[phi].sub.F] (u) := [[phi].sub.F]([u.sub.+]) - [[phi].sub.F]([u.sup.-]) for all u [member of] E

(see [BuL, [section] 3, Lemma 7]). Clearly [[phi].sub.F] [member of] [E.sup.~] and for f [member of] E(X) we have

[[phi].sub.F]([??]) = [absolute value of F](f) for all f [member of] E(X).

Now we recall the concept of solidness in E[(X).sup.~] (see [[N.sub.1], [section] 2], [[N.sub.2]]). For [F.sub.1], [F.sub.2] [member of] E[(X).sup.~] we will write [absolute value of [F.sub.1]] [less than or equal to] [absolute value of [F.sub.2]] whenever [absolute value of [F.sub.1]](f)] [less than or equal to] [absolute value of [F.sub.2]](f) for all f [member of] E(X). A subset A of E[(X).sup.~] is said to be solid whenever [absolute value of [F.sub.1]] [less than or equal to] [absolute value of [F.sub.2]] with [F.sub.1] [member of] [(X).sup.~] and [F.sub.2] [member of] A imply [F.sub.1] [member of] A. A linear subspace I of E[(X).sup.~] will be called an ideal of E[(X).sup.~] whenever I is solid.

An order bounded linear functional F on E(X) is said to be smooth whenever for a net ([f.sub.[alpha]]) in E(X), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in E implies F([f.sub.[alpha]]) [right arrow] 0 (see [BuL, [section] 3, Definition 2], [[N.sub.1]], [[N.sub.2]]). (Note that Bukhvalov and Lozanovskii [BuL] use the term "integral" and in [[N.sub.1]], [[N.sub.2]] we use the term "order continuous"). The set consisting of all smooth functionals on E(X) will be denoted by E[(X).sup.~.sub.n]. Note that E[(X).sup.~.sub.n] separates the points of E(X) because we assume that supp E' = [OMEGA].

A subset H of E(X) is said to be solid whenever [[??].sub.1] [less than or equal to] [[??].sub.2] and [f.sub.1] [member of] E(X), [f.sub.2] [member of] H imply [f.sub.1] [member of] H. A linear topology [tau] on E(X) is said to be locally solid if it has a local base at zero consisting of solid sets. A locally solid topology [tau] on E(X) is said to be a Lebesgue topology whenever for a net ([f.sub.[alpha]]] in E(X), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in E implies [f.sub.[alpha]] [right arrow] 0 for [tau] (see [[N.sub.3], Definition 2.2]).

It is known that a Banach space X is an Asplund space if and only if [X.sup.*] has the Radon-Nikodym property (see [DU, p. 213]).

The following theorem will be of importance (see [Ns, Theorems 1.2 and 4.1]):

Theorem 2.1. Assume that X is an Asplund space. Then the Mackey topology [tau](E(X), E[(X).sup.~.sub.n]) is a locally convex-solid Lebesgue topology.

Recall that a functional F [member of] E[(X).sup.~] is said to be singular if there exists an ideal M of E with supp M = [OMEGA] and such that F(f) = 0 for all f [member of] M(X). The set consisting of all singular functionals on E(X) will be denoted by E[(X).sup.~.sub.s] and called the singular dual of E(X) (see [BuL, [section]3, Definition 2]).

It is known that E[(X).sup.~.sub.n] and E[(X).sup.~.sub.s] are ideals of E[(X).sup.~] (see [[N.sub.1]]).

Due to Bukhvalov and Lozanovski (see [BuL, [section]3, Theorem 2]) we have the following Yosida-Hewitt type decomposition of E[(X).sup.~].

Theorem 2.2. The following decomposition of E(X)~ holds:

(1.1) E[(X).sup.~] = E[(X).sup.~.sub.n] [direct sum] E[(X).sup.~.sub.s]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whenever F = [F.sub.1] + [F.sub.2] with [F.sub.1] [member of] E[(X).sup.~.sub.n], [F.sub.2] [member of] E[(X).sup.~.sub.s]. Moreover, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One can note that E[(X).sup.~.sub.n] = E[(X).sup.~] if and only if [E.sup.~.sub.n] = [E.sup.~].

In view of (1.1) we have linear projections : [P.sub.k] : E[(X).sup.~] [right arrow] E[(X).sup.~] (k = 1,2) defined by [P.sub.k](F) = [F.sub.k]. Note that for F [member of] E[(X).sup.~] and every f [member of] E(X) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 2.3. For a linear operator T : E(X) [right arrow] Y the following statements are equivalent:

(i) [y.sup.*] [omicron] T [member of] E[(X).sup.~.sub.n] for every [y.sup.*] [member of] [Y.sup.*].

(ii) T is ([sigma](E(X), E[(X).sup.~.sub.n]), [sigma](Y, [Y.sup.*]))-continuous.

(iii) T is ([tau](E(X), E[(X).sup.~.sub.n]), [parallel] x [parallel])-continuous.

Proof. (i) [??] (ii) See [[AB.sub.2], Theorem 9.26]; (ii) [??] (iii) See [W, Corollary 11-1-3, Corollary 11-2-6].

Following [BBuY] we define smooth and singular operators on E(X).

Definition 2.1. (i) An order bounded linear operator T : E(X) [right arrow] Y is said to be smooth if for a net ([f.sub.[alpha]]) in E(X), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in E implies [[parallel]T([f.sub.[alpha]])[parallel].sub.Y] [right arrow] 0.

(ii) An order bounded linear operator T : E(X) [right arrow] Y is said to be singular if there exists an ideal M of E with supp M = [OMEGA] such that T(f) = 0 for all f [member of] M(X).

(iii) An order bounded linear operator T : E(X) [right arrow] Y is said to be weakly singular if [y.sup.*] [omicron] T [member of] E[(X).sup.~.sub.s] for every [y.sup.*] [member of] [Y.sup.*].

The following theorem gives a characterization of smooth operators T : E(X) [right arrow] Y when X is an Asplund space.

Theorem 2.4. Assume that X is an Asplund space. Then for a linear operator T : E(X) [right arrow] Y the following statements are equivalent:

(i) T is smooth.

(ii) [y.sup.*] [omicron] T [member of] E[(X).sup.~.sub.n] for every [y.sup.*] [member of] [Y.sup.*].

(iii) T is ([tau](E(X), E[(X).sup.~.sub.n]), [[parallel] x [parallel].sub.Y])-continuous.

Proof. (i)=>(ii) It is obvious. (ii) [??] (iii) See Proposition 2.3. (iii)=>(i) Clearly, because [tau](E(X), E[(X).sup.~.sub.n]) is a Lebesgue topology (see Theorem 2.1).

We will need the following lemma.

Lemma 2.5. Assume that [absolute value of F] [less than or equal to] [absolute value of G], where F, G [member of] E[(X).sup.~]. Then [absolute value of [P.sub.k](F)] [less than or equal to] [absolute value of [P.sub.k](G)] for k = 1,2.

Proof. We have F = [F.sub.1] + [F.sub.2], G = [G.sub.1] + [G.sub.2], where [F.sub.1], [G.sub.1] [member of] E[(X).sup.~.sub.n], [F.sub.2], [G.sub.2] [member of] E[(X).sup.~.sub.n] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see (1.1)). Let u [member of] [E.sup.+] and [x.sub.0] [member of] [S.sub.X] be fixed. Then

[[phi].sub.F](u) = [absolute value of F](u [cross product] [x.sub.0]) [less than or equal to] [absolute value of G](u [cross product] [x.sub.0]) = [[phi].sub.g](u).

Since the order projections of [E.sup.~] onto [E.sup.~.sub.n] and [E.sup.~.sub.s] are positive operators, for f [member of] E(X) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For a linear functional V on E[(X).sup.~] let us put:

[absolute value of V](F) = sup{[absolute value of V(G)] : G [member of] E[(X).sup.~], [absolute value of G] [less than or equal to] [absolute value of F]} for F [member of] E[(X).sup.~].

The set

[(E[(X).sup.~]).sup.~] = {V [member of] [(E[(X).sup.~]).sup.#] : [absolute value of V](F) < [infinity] for all F [member of] E[(X).sup.~]}

will be a called the order dual of E[(X).sup.~] (see [[N.sub.2]]) (here [(E[(X).sup.~]).sup.#] denotes the algebraic dual of E[(X).sup.~]).

For [V.sub.1], [V.sub.2] [member of] (E[(X).sup.~])~ we will write [absolute value of [V.sub.1]] [less than or equal to] [absolute value of [V.sub.2]] whenever [absolute value of [V.sub.1]](F) [less than or equal to] [absolute value of [V.sub.2]](F) for all F [member of] E[(X).sup.~]. A subset K of [(E(X)~).sup.~] is said to be solid whenever [absolute value of [V.sub.1]] < [absolute value of [V.sub.2]] with [absolute value of [V.sub.1]] [member of] [(E[(X).sup.~]).sup.~], [V.sub.2] [member of] K imply [V.sub.1] [member of] K. A linear subspace L of [(E[(X).sup.~]).sup.~] is called an ideal if L is a solid subset of [(E[(X).sup.~]).sup.~].

For each f [member of] E(X) let us put

[[pi].sub.f](F) = F(f) for all F [member of] E[(X).sup.~].

One can show (see [[N.sub.2], [section]1]) that for [member of] E(X),

[absolute value of [[pi].sub.f]](F) = [absolute value of F](f) for F [member of] E[(X).sup.~] and that [[pi].sub.f] [member of] [(E[(X).sup.~]).sup.~].

Thus we have a natural embedding [pi] : E(X) [contains as member] f [??] [[pi].sub.f] [member of] [(E[(X).sup.~]).sup.~].

Denote by E[(X).sub.0] the ideal of [(E[(X).sup.~]).sup.~] generated by the set [pi](E(X)), i.e., E[(X).sub.0] is the smallest ideal of [(E[(X).sup.~]).sup.~] containing [pi](E(X)). One can show that (see [[N.sub.2], Theorem 3.2]):

E[(X).sub.0] = {V [member of] [(E[(X).sup.~]).sup.~] : [absolute value of V] [less than or equal to] [absolute value of [pi].sub.f]] for some f [member of] E(X)}.

Let

[P.sup.~.sub.k] : [(E[(X).sup.~]).sup.#] [right arrow] (E(X)~).sup.#]

stand for the conjugate of [P.sub.k] (k = 1,2) defined by

[P.sup.~.sub.k](V)(F) = V([P.sub.k](F)) for V [member of] [(E[(X).sup.~]).sup.#] and F [member of] E[(X).sup.~].

Observe that

[P.sup.~.sub.k]([(E[(X).sup.~]).sup.~]) [subset] [(E[(X).sup.~]).sup.~].

Indeed, let V [member of] ([(E[(X).sup.~]).sup.~]. Then by making use of Lemma 2.5 we have for F [member of] E[(X).sup.~],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, in particular, we get:

Corollary 2.6. Let f [member of] E(X). Then for every F [member of] E[(X).sup.~] we have

[absolute value of [P.sup.~.sub.k]([[pi].sub.f])](F) [less than or equal to] [absolute value of [[pi].sub.f]]([P.sub.k](F)) [less than or equal to] [absolute value of [[pi].sub.f](F),

and hence [P.sup.~.sub.k]([[pi].sub.f]) [member of] E[(X).sub.0] (k = 1,2).

3 A Yosida-Hewitt type decomposition for order-weakly compact operators

In this section we derive Yosida-Hewitt type decompositions for order-weakly compact operators T : E(X) [right arrow] Y.

Assume now that T : E(X) [right arrow] Y is an order bounded operator, i.e., T is ([tau](E(X), E[(X).sup.~]), [[parallel] x [parallel].sub.Y])-continuous. It follows that [y.sup.*] [omicron] T [member of] E[(X).sup.~] for every [y.sup.*] [member of] [Y.sup.*]. Then we can consider the linear mappings (see [[N.sub.6]]):

[T.sup.~] : [Y.sup.*] [right arrow] E[(X).sup.~]

defined by

[T.sup.~]([y.sup.*])(f) = [y.sup.*](T(f)) for [y.sup.*] [member of] Y and all f [member of] E(X),

and

[T.sup.~~] : E[(X).sub.0] [right arrow] [Y.sup.**]

defined by

[T.sup.~~](V)([y.sup.*]) = V([T.sup.~]([y.sup.*])) for V [member of] E[(X).sub.0] and all [y.sup.*] [member of] [Y.sup.*].

The map [T.sup.~~] is ([sigma](E[(X).sub.0], E[(X).sup.~]), [sigma]([Y.sup.**], [Y.sup.*]))-continuous.

Let i : Y [contains as member] y [??] [i.sub.y] [member of] [Y.sup.**] stand for the canonical isometry, i.e., [i.sub.y]([y.sup.*]) = [y.sup.*](y) for [y.sup.*] [member of] [Y.sup.*]. Moreover, let j : i(Y) [right arrow] Y stand for the left inverse of i, i.e., j [omicron] i = [id.sub.Y]. Then [T.sup.~~] [omicron] [pi] = i [omicron] T.

The following characterization of order-weakly compact operators T: E(X) [right arrow] Y will be of importance.

Theorem 3.1 (see [[N.sub.5], Theorem 2.3]). For an order-bounded operator T : E(X) [right arrow] Y the following statements are equivalent:

(i) T is order-weakly compact.

(ii) [T.sup.~~](E[(X).sub.0]) [subset] i(Y).

For f [member of] E(X) let us set

[I.sub.f] = {V [member of] E[(X).sub.0]: [absolute value of V] [less than or equal to] [absolute value of [[pi].sub.f]]}.

The following property of [I.sub.f] will be needed.

Theorem 3.2. For f [member of] E(X) the set [I.sub.f] is [sigma](E[(X).sub.0], E[(X).sup.~])-compact in E[(X).sub.0].

Proof. Clearly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We shall show that [I.sub.f] is a totally bounded and closed set in [((E[(X).sup.~]).sup.#], [sigma][((E[(X).sup.~]).sup.#], E[(X).sup.~])). In fact, let F [member of] E[(X).sup.~]. Then for each V [member of] [I.sub.f] we have

[absolute value of V(F)] [less than or equal to] [absolute value of V](F) [less than or equal to] [absolute value of [[pi].sub.f](F) = [absolute value of F](f) < [infinity].

This means that [I.sub.f] is bounded for [sigma]([(E[(X).sup.~]).sup.*], E[(X).sup.~]), so by [KA, Lemma 3.3.5] it is totally bounded in [(E[(X).sup.~]).sup.#], [sigma]([(E[(X).sup.~]).sup.#], E[(X).sup.~])).

To see that [I.sub.f] is closed in ([(E[(X).sup.~]).sup.#], [sigma] ([(E[(X).sup.~]).sup.#], E[(X).sup.~])), assume that [V.sub.[alpha]] [right arrow] V for [sigma] ([(E[(X).sup.~]).sup.#], E[(X).sup.~]), where (K) is a net in [I.sub.f] and V [member of] [(E[(X).sup.~]).sup.#]. It is enough to show that [absolute value of V] [less than or equal to] [[pi].sub.f], i.e., [absolute value of V](F) [less than or equal to] [absolute value of [[pi].sub.f]](F) = [absolute value of F](f) for each F [member of] E[(X).sup.~]. In fact, let F [member of] E(X)~ and e > 0 be given. Let G [member of] E[(X).sup.~] and [absolute value of G] [less than or equal to] [absolute value of F]. Since [V.sub.[alpha]](G) [right arrow] V(G), there exists [[alpha].sub.0] such that for [alpha] [greater than or equal to] [[alpha].sub.0] we get

[absolute value of (G)] [less than or equal to] [absolute value of [V.sub.[alpha](G)] + [epsilon] [less than or equal to] [absolute value of [V.sub.[alpha](G)] + [epsilon] [less than or equal to] [absolute value of [[pi].sub.f]](G) + [epsilon] [less than or equal to] [absolute value of [[pi].sub.f]](F) + [epsilon].

It follows that [absolute value of V](F) [less than or equal to] [absolute value of [[pi].sub.f](F), so [absolute value of V] [less than or equal to] [[pi].sub.f], as desired.

Since the space [((E[(X).sup.~]).sup.#], [sigma] [((E[(X).sup.~]).sup.#], E[(X).sup.~])) is complete (see [KA, Lemma 3.3.4]), the set [I.sub.f] is complete for [sigma] [((E[(X).sup.~]).sup.#], [((E[(X).sup.~]), so we can conclude that If is compact for [sigma] [((E[(X).sup.~]).sup.#], [((E[(X).sup.~]) (see [KA, Theorem 3.1.4]). It follows that If is also [sigma](E[(X).sub.0], [((E[(X).sup.~])-compact.

Now we are in position to prove our main result.

Theorem 3.3. Let T : E(X) [right arrow] Y be an order-weakly compact operator. Then T can be uniquely decomposed as T = [T.sub.1] + [T.sub.2], where [T.sub.1], [T.sub.2] are order-weakly compact operators, [T.sub.1] is ([tau](E(X), E[(X).sup.~.sub.n]), [[parallel] x [parallel].sub.Y])-continuous and [T.sub.2] is weakly singular.

Proof. In view of Corollary 2.6, [P.sup.~.sub.f]{[[pi].sub.f]) [member of] E[(X).sub.0] (k = 1,2). Hence by Theorem 3.1, [T.sup.~~]([P.sup.~.sub.k]([[pi].sub.f])) [member of] i(Y), and we can define linear mappings:

[T.sub.k] = j [omicron] [T.sup.~~] [omicron] [P.sup.~.sub.k] [omicron] [pi] : E(X) [right arrow] Y.

Then for [y.sup.*] [member of] [Y.sup.*] and f [member of] E(X) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and this means that [T.sub.1] is ([tau](E(X), E[(X).sup.~.sub.n]), [[parallel] x [parallel].sub.Y])-continuous (see Proposition 2.2) and [T.sub.2] is weakly singular (see Definition 2.1). Moreover, for every [y.sup.*] [member of] [Y.sup.*] and f [member of] E(X) we have

[y.sup.*]([T.sub.1](f) + [T.sub.2](f)) = [P.sub.1]([y.sup.*] [omicron] T)(f) + [P.sub.2]([y.sup.*] [omicron] T)(f) = [y.sup.*](T(f)),

so T(f) = [T.sub.1](f) + [T.sub.2](f). The uniqueness of the decomposition T = [T.sub.1] + [T.sub.2] follows from the uniqueness of the decomposition [y.sup.*] [omicron] T = [y.sup.*] [omicron] [T.sub.1] + [y.sup.*] [omicron] [T.sub.2] for each [y.sup.*] [member of] [Y.sup.*] (see (1.1)).

Now we shall show that [T.sub.k] : E(X) [right arrow] Y are order-weakly compact operators. Indeed, let u [member of] [E.sup.+] and [D.sub.u] = {h [member of] E(X) : [??] [less than or equal to] u}. In view of Corollary 2.6 for h [member of] [D.sub.u] and a fixed [x.sub.0] [member of] [S.sub.X] we get for F [member of] E[(X).sup.~]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. According to Theorem 3.2 the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [sigma](E[(X).sub.0], E[(X).sup.~])-compact in E[(X).sub.0], and this means that {[P.sup.~.sub.k]([[pi].sub.h]) : h [member of] [D.sub.u]} is a relatively [sigma](E[(X).sub.0], E[(X).sup.~])-compact subset of E[(X).sub.0].

Since [T.sup.~~](E[(X).sub.0]) [subset] i(Y) [subset] [Y.sup.**] and [T.sup.~~] is [sigma](E[(X).sub.0], E[(X).sup.~]), [sigma]([Y.sup.**], [Y.sup.*])) continuous, the set {[T.sup.~~]([P.sup.~.sub.k]([[pi].sub.h])) : h [member of] [D.sub.u]} is relatively [sigma]([Y.sup.**], [Y.sup.*])-compact in [Y.sup.**]. But the mapping j is ([sigma](i(Y), [Y.sup.*]), [sigma](Y, [Y.sup.*]))-continuous, so the set [T.sub.k]([D.sub.u]) = {j([T.sup.~~]([P.sup.~.sub.k]([[pi].sub.h]))) : h [member of] [D.sub.u]} is relatively [sigma](Y, [Y.sup.*])-compact in Y.

Using Theorems 2.4 and 3.3 we obtain the following Yosida-Hewitt type decomposition for order-weakly compact operators T : E(X) [right arrow] Y.

Theorem 3.4. Let T : E(X) -> Y be an order weakly compact operator. Assume that X is an Asplund space. Then T can be uniquely decomposed as T = [T.sub.1] + [T.sub.2], where [T.sub.1], [T.sub.2] are order-weakly compact, [T.sub.1] is smooth and [T.sub.2] is weakly singular.

From now on we assume that (E, [[parallel] x [parallel].sub.E]) is a Banach function space. Then the space E(X) provided with the norm [[parallel] f [parallel].sub.E(X)] := [[parallel] [??] [parallel].sub.E] is a Banach space and is usually called a Kothe-Bochner function space. Then the Mackey topology [tau](E(X), E[(X).sup.~]) coincides with the [[parallel] x [parallel].sub.E(X)]-norm topology and a linear operator T : E(X) [right arrow] Y is order bounded if and only if T is ([[parallel] x [parallel].sub.E(X)], [[parallel] x [parallel].sub.Y])-continuous (see [[N.sub.4], Theorem 2.3]). Let

[E.sub.a] = {u [member of] E : [absolute value of u] [greater than or equal to] [u.sub.n] [down arrow] 0 in E implies [[parallel] [u.sub.n] [parallel].sub.E] [right arrow] 0}.

It is well known that [E.sub.a] is [[parallel] x [parallel].sub.E]--closed ideal of E and [E.sub.a] = E if and only if [[parallel] x [parallel].sub.E] is order continuous.

We will need the following useful characterization of singular operators on Kothe-Bochner function spaces (see [[N.sub.7], Proposition 1.4]).

Proposition 3.5. Assume that (E, [[parallel] x [parallel].sub.E]) is a Banach function space with supp [E.sub.a] = [OMEGA]. Then for a ([[parallel] x [parallel].sub.E(X)], [[parallel] x [parallel].sub.Y])-continuous linear operator T : E(X) [right arrow] Y the following statements are equivalent:

(i) T is singular.

(ii) T is weakly singular.

(iii) T(f) = 0 for all f [member of] [E.sub.a](X).

Combining Theorem 3.4 with Proposition 3.5 we are ready to state a Yosida-Hewitt type decomposition for order-weakly compact operators acting from Kothe-Bochner function spaces E(X) to Banach spaces.

Theorem 3.6. Assume that (E, [[parallel] x [parallel].sub.E]) is a Banach function space with supp [E.sub.a] = [OMEGA] and X is an Asplund space. Let T : E(X) [right arrow] Y be an order-weakly compact operator. Then T can be uniquely decomposed as T = [T.sub.1] + [T.sub.2], where [T.sub.1], [T.sub.2] are order-weakly compact operators, [T.sub.1] is smooth and [T.sub.2] is singular.

Acknowledgements. The author is very grateful to the referee for many useful corrections and suggestions which have improved the paper.

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Received by the editors February 2009--In revised form in March 2010.

Communicated by F. Bastin.

2000 Mathematics Subject Classification : 47B38, 47B07, 46E40, 46A20.

Faculty of Mathematics, Computer Science and Econometrics

University of Zielona Gora

ul. Szafrana 4A, 65-516 Zielona Gora, Poland

e-mail: M.Nowak@wmie.uz.zgora.pl

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Author: | Nowak, Marian |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Date: | May 1, 2011 |

Words: | 5269 |

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