Printer Friendly

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

1 Introduction and preliminaries

The 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.

References

[[AB.sub.1]] Aliprantis, C.D. and O. Burkinshaw, On positive order continuous operators, Indag. Math., 45 (1983), 1-6.

[[AB.sub.2]] Aliprantis, C.D. and O. Burkinshaw, Positive Operators, Academic Press, Orlando, San Diego, New York, Tokyo, 1985.

[BBuY] Basile, A., Bukhvalov, A.V. and M.Ya. Yakubson, The generalized Yosida-Hewitt theorem, Math. Proc. Camb. Phil. Soc., 116 (1994), 527-533.

[BBu] Basile, A. and A.V. Bukhvalov, On a unifying approach to decomposition theorems Yosida-Hewitt type, Ann. Math. Pura Appl., 173 (4), (1997), 107125.

[BuL] Bukhvalov, A.V. and G.Ya. Lozanovskii, On sets closed in measure in spaces of measurable functions, Trans. Moscow Math. Soc., 2, (1978), 127-148.

[CM] Cembranos, P. and J. Mendoza, Banach spaces of vector-valued functions, Lectures Notes in Math., 1676, Springer Verlag, Berlin, Heidelberg, 1997.

[DU] Diestel, J. and J.J. Uhl, Vector Measures, Amer. Math. Soc., Math. Surveys, no. 15, Providence, Rhode Island 1977.

[D] Dodds, P.G., o-weakly compact mappings of Riesz spaces, Trans. Amer. Math. Soc., 214 (1975), 389-402.

[E] Eldik van, P., The integral component of an order bounded transformation, Quastiones Math., 1 (1976), 135-144.

[KA] Kantorovitch, L.V. and A.V. Akilov, Functional Analysis, Pergamon Press, Oxford-Elmsford, N.Y.,1982.

[KM] Kusraev, A.G. and S.A. Malyugin, On the order continuous component of a majorized operator, Siberian J. Math., 28 (1988), no. 4, 617-627.

[L] Lin Pei-Kee, Kothe-Bochner Function Spaces, Birkhauser Verlag, Boston, Besel, Berlin, 2003.

[[N.sub.1]] Nowak, M., Duality theory of vector valued function spaces I, Comment. Math., Prace Mat., 37 (1997), 195-215.

[[N.sub.2]] Nowak, M., Duality theory of vector valued function spaces II, Comment. Math., Prace Mat., 37 (1997), 217-230.

[[N.sub.3]] Nowak, M., Lebesgue topologies on vector-valued function spaces, Math. Japonica 52, no. 2 (2000), 171-182.

[[N.sub.4]] Nowak, M., Order bounded operators from vector-valued function spaces to Banach spaces, Proc. Conf. Function Spaces VII, Poznan 2003, Banach Center Publ., 68 (2005), 109-114.

[[N.sub.5]] Nowak, M., Order-weakly compact operators from vector-valued function spaces to Banach spaces, Proc. Amer. Math. Soc., 135, no. 9 (2007), 2803-2809.

[[N.sub.6]] Nowak, M., Linear operators on vector-valued function spaces with Mackey topologies, J. Convex Analysis, 15 no. 1 (2008), 165-178.

[[N.sub.7]] Nowak, M., Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures, J. Math. Anal. Appl., 336 no. 1 (2007), 93-100.

[S] Schep, A.R., Order continuous components of operators and measures, Indag. Math., 40, no. 1 (1978), 110-117.

[W] Wilansky, A., Modern methods in Topological Vector-Spaces, Mc Graw Hill, 1978.

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
COPYRIGHT 2011 Belgian Mathematical Society
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2011 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Nowak, Marian
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Article Type:Report
Date:May 1, 2011
Words:5269
Previous Article:A subordination result with Salagean-type certain analytic functions of complex order.
Next Article:Constant angle surfaces in Minkowski space.
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters