SOME CHANGE OF VARIABLE FORMULAS IN INTEGRAL REPRESENTATION THEORY. 1. Introduction

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1 First Prev Next Last Go Back Full cree Close Quit OM CHANG OF VARIABL FORMULA IN INTGRAL RPRNTATION THORY L. MZIANI Abstract. Let X, Y be Baach spaces ad let us deote by C(, X) the space of all X-valued cotiuous fuctios o the compact Hausdorff space, equipped with the uiform orm. We shall write C(, X) = C() if X = R or C. Now, cosider a bouded liear operator T : C(, X) Y ad assume that, due to the effect of a chage of variable performed by a bouded operator V : C(, X) C(), the operator T takes the product form T = θ V, with θ : C() Y liear ad bouded. I this paper, we prove some itegral formulas givig the represetig measure of the operator T, which appeared as a essetial object i itegral represetatio theory. This is made by meas of the represetig measure of the operator θ which is geerally easier. ssetially the estimatios are of the Rado-Nikodym type ad precise formulas are stated for weakly compact ad uclear operators. 1. Itroductio Let be a compact Hausdorff space ad B the σ-field of the Borel sets of. I all what follows, X ad Y will be fixed Baach spaces ad we cosider the Baach space C (, X) of all X-valued cotiuous fuctios o, with the uiform orm; we write C (, X) = C () whe X = R or C. I this work, we will be cocered with the itegral aalysis of bouded operators T : C (, X) Y, takig the form: (1.1) T = θ V Received May 9, Mathematics ubject Classificatio. Primary 28C05; ecodary 46G10. Key words ad phrases. Chage of variable i Bouded Operators, Vector measures, Weakly compact & Nuclear Operators.

2 First Prev Next Last Go Back Full cree Close Quit due to the effect of a chage of variable performed by a bouded operator V : C (, X) C (); θ beig a bouded operator o C () with values ito Y. Whe the operators T ad V are give, we will show how to get the operator θ : C () Y, satisfyig the product form (1.1). The we determie the structure of the additive operator valued measure G : B L (X, Y ) attached to the operator T via the itegral represetatio: (1.2) f C (, X), T f = f dg. Accordig to the Theorem of Diculeau [2, 19], L (X, Y ) is the Baach space of all bouded operators from X ito the secod cojugate space Y of Y. I doig the computatios, we shall make use of the itegral form (1.3) g C (), θg = g dµ of the operator θ, give by Bartle-Duford-chwartz, [3, VI-7]; i this cotext µ is a vector set fuctio o B with values i Y (resp. a vector measure with values i Y, if θ is weakly compact). As we will see, the relatios betwee G ad µ are, i some sese, of the Rado-Nikodym type. We shall compute explicitly the derivatives arisig from these relatios. The most precise results about the vector measure G are obtaied for weakly compact ad uclear operators T. The paper is orgaized as follows. I ectio 2 we will make precise the chage of variable V : C (, X) C () leadig to the product form (1.1). Also we recall some facts from itegral represetatio theory givig (1.2) ad (1.3). I ectio 3 we give a geeral estimatio formula for the measure G by meas of the set fuctio µ. We examie i sectio 4 the case of weakly compact operators T, which allows a improvemet of the estimatio made i ectio 3. We cosider uclear operators T i ectio 5. If T takes the form (1.1) by a chage of variable V : C (, X) C (), we show how we ca recover the uclear property for the compoet θ. The we prove that the measure G is a Bocher itegral with respect to a bouded scalar measure. A simple example is give i ectio 6, where all computatios of ectios 2 5 are performed explicitly. Fially, ectio 7 is iteded to a remark about aother estimatio of G made elsewhere [5, 5].

3 First Prev Next Last Go Back Full cree Close Quit 2. The chage of variable V : C (, X) C (). I all what follows, we will always assume that C (, X) is mapped oto C () by the operator (2.1) C (, X) is mapped oto C () by the operator V. We eed this hypothesis i costructig the compoet θ : C () Y as a bouded operator givig the product form T = θ V. The operator V i (2.1) may be cosidered as performig a chage of variable from the space C (, X) to the space C (). Oe usefull fact about V is: Propositio 1. There exists a costat K > 0, such that for every h C (), there is a solutio f C (, X) of h = V f, satisfyig f K h. Proof. ice V is oto, the by the ope mappig Theorem, the ope uit ball B of C (, X) maps oto a set V B which cotais some relative ope ball {u V B : u < α}, with α > 0. Thus, for 0 h V C (, X) = C (), the vector α h 2 h g 2 h is the image uder V of a vector g, with g < 1. Hece if we put f = α, we have V f = h ad f 2 α h, which proves the propositio with K = 2 α. The effect of a chage of variable V : C (, X) C () is give by: Theorem 1. A bouded operator T : C (, X) Y factors as T = θ V, where θ : C () Y is a bouded operator, if ad oly if the followig coditio is satisfied: (2.2) Ker V Ker T Proof. The ecessity of the coditio is clear. To see that it is sufficiet, we first proceed to the costructio of θ. Let h C (), the citig (2.1) gives a f C (, X) such that h = V f; let us put θh = T f. The θ is a well defied mappig; for, if V f 1 = V f 2 = h, where f 1, f 2 C (, X), the we have f 1 f 2 Ker V which implies f 1 f 2 Ker T by (2.2); so T f 1 = T f 2. It is clear that θ is liear ad that we have T f = θ V f, for all f C (, X).

4 First Prev Next Last Go Back Full cree Close Quit We must show that θ is bouded. By Propositio 1 there exists K > 0 such that for every h C () we ca choose a solutio f of h = V f so that f K h. Therefore we have θh = T f T f T K h, which gives the boudedess of θ. Remark 1. It is oteworthy that we may relax the assumptio (2.1) if we require from V to be of closed rage. I this case we still have the validity of both Propositio 1 ad Theorem 1, but with θ defied ad bouded o the rage of V. Before statig the Theorems we eed i the cotext of vector itegratio, let us put some prelimiaries ad facts for later use. Defiitio 1. Let G : B L (X, Y ) be a fiitely additive vector measure o B with values i the Baach space L (X, Y ). For each y Y, let us defie the set fuctio G y : B X by: (2.3) B, x X : G y () (x) = y G () (x) that is, the fuctioal G () (x) of Y applied to the vector y Y. The it is a simple fact that G y is for each y Y a fiitely additive X -valued measure o B. The family of measures {G y, y Y } iduces i tur a family of scalar fiitely additive measures { My x : x X, y Y } defied by: (2.4) B, x X, y Y : My x () = G y () (x). Let us recall also the otios of variatio ad semivariatio of a measure: Defiitio 2. Let Z be a Baach space ad µ : B Z a vector measure (ote that µ may be scalar). The (a) The variatio of µ is the set fuctio v(µ, ) of B i [0, + ] defied by: (2.5) B : v(µ, ) = sup π µ(a) the sup is over all fiite partitios π of by sets i B. Call v(µ, ) = v(µ), the variatio of µ. A π

5 First Prev Next Last Go Back Full cree Close Quit (b) The semivariatio of µ is the set fuctio µ : B [0, + ] defied by the formula: (2.6) B : µ () = sup {v(z µ, ) : z Z, z 1} ote that z µ is scalar for each z Z. Defiitio 3. We say that a vector measure µ : B Z is regular if for each B ad ε > 0 there exist a ope set O ad a compact set K such that, K O ad µ (O \ K) < ε. If the measure µ is scalar this iequality may be replaced by v(µ, O \ K) < ε [1, Chapter 1] for all relatios betwee the set fuctios v(µ, ) ad µ ). With the igrediets above, we have: Propositio 2. uppose that the measure G y is bouded ad regular for some y Y the we have (i) G y is coutably additive. (ii) The scalar measures My x are coutably additive ad regular for each x X. Proof. Let B ad ε > 0, the there exist a ope set O ad a compact set K such that, K O ad G y (O \ K) < ε. ice G y is X -valued, we have G y (O \ K) = sup {v(x G y, O \ K) : x X, x 1} < ε by (2.6). This implies that the family of scalar set fuctios {x G y : x X, x 1} is uiformly regular; sice they are additive, we deduce, by the Theorem III.5.13 i [3], that x G y is coutably additive for each x X, x 1 ad the also for all x X. Cosequetly G y is coutably additive by the Orlicz-Pettis Theorem. To see part (ii), let γ : X X deote the caoical isomorphism of X ito X, ad let us observe that My x = γ (x) G y, by formula (2.4); therefore we deduce that the scalar measure M y x is coutably additive ad regular for each x X.

6 First Prev Next Last Go Back Full cree Close Quit Now we tur to the itegral represetatio Theorems we shall eed i the sequel. Theorem 2. Let T : C (, X) Y be a liear bouded operator. The there exists a uique additive operator valued measure G : B L (X, Y ) such that: (2.7) T f = f (s) dg (we call G the represetig measure of the operator T ). Moreover, for each y Y, G y is a regular coutably additive bouded X -valued measure ad we have (2.8) T y = G y where T is the adjoit of T ad where the idetificatio, betwee the dual space C (, X) ad the Baach space rcab (B, X ) of X -valued measures o B is used. Because of the equatio (2.8) we shall call the family of measures {G y, y Y }, the adjoit family of G or of T. For the proof see referece [2, 19]. Theorem 3. Let θ : C() Y be a bouded liear operator. The there exists a uique set fuctio µ : B Y such that (a) µ( )y is a regular coutably additive scalar measure o B for all y Y (i symbols µ( )y rca()). (b) y θf = f (s) dµ (s) y for all y Y ad f C(). We call µ the represetig measure of θ. Moreover, if the operator θ is weakly compact, the µ is a true coutably additive measure with values i Y such that (a ) y µ is a regular scalar measure for all y Y. (b ) θf = f (s) dµ (s) for all f C().

7 First Prev Next Last Go Back Full cree Close Quit O the other had, if θ : Y C () is the adjoit of θ the we have θ y = y µ for all y Y. For the proof see [3, VI.7.2 ad VI.7.3]. 3. Geeral estimatio of the represetig measures Let T : C (, X) Y ad V : C (, X) C () be bouded operators ad suppose that T factors as T = θ V, where θ : C() Y is bouded. I this sectio, we will prove a geeral formula betwee the represetig measures G ad µ of the operators T ad θ. We will see that the resultig relatios betwee G ad µ are of the Rado-Nikodym type ad we will give the expressio of the derivatives by meas of the operator V. To make the estimatio tractable we shall impose o the operator V the followig coditio (3.1) g C(), h C (, X) : V (g h) = g V (h). I the computatios below, we eed coditio (3.1) to be satisfied oly for the costat fuctios h C (, X). Here is a example of a o trivial bouded V : C (, X) C () satisfyig (3.1): xample 1. Let K : R be a cotiuous fuctio ad let µ be a measure with bouded variatio o B. Let us cosider the operator φ : C() C(), defied by: φ (g) (s) = K (s, t) g (t) dµ (t). The fact that K is cotiuous ad µ of bouded variatio makes it easy to prove that φ (g) is i C(). Now take X = C() ad defie V : C(, X) C(), by h C(, X), V (h) (r) = φ (h r ) (r), for r. Let us ote that the value h r, of the fuctio h at the poit r, is i C() because h C(, X), ad X = C(). Note also, from the defiitio of φ, that we have V (h) (r) = K (r, t) h r (t) dµ (t). It is ot difficult to show that the fuctio r V (h) (r) is cotiuous ad that V : C(, X) C() is a liear bouded operator with V M K v (µ), where M K = sup { K (s, t), (s, t) }. We prove that V satisfies (3.1).

8 First Prev Next Last Go Back Full cree Close Quit Let g C(), h C(, X), the we have V (g h)(r) = K(r, t)g(r) h r (t) dµ(t) = g (r) K(r, t)h r (t) dµ (t) = g (r) V (h) (r). (For a other example of operator satisfyig (3.1), see ectio 6 below.) We ow state ad prove the geeral estimatio Theorem. Recall the measures G y, My x i (2.3) ad (2.4), ad µ( )y i Theorem 3(a). Theorem 4. Uder (2.1), (2.2), (3.1), the operator T factors as T = θ V ad we have (3.2) G y () (x) = V (c x ) (t) dµ (t) y. for all B, y Y ad x X, where c x C (, X) is the costat fuctio X give by c x (t) x, x beig fixed i X. I other words the measure My x is absolutely cotiuous with respect to µ( )y, with Rado-Nikodym derivatives give by d M x y d µ( )y = V (c x ), so we may write (3.2) as d My x = V (c x) d µ( )y. Proof. First let us apply the itegral (2.7) to the fuctio f C (, X) of the form f (t) = g (t) c x (t), with g C () ad x fixed i X. We obtai T g c x = g c x dg, ad for y Y y T g c x = g c x dg y = g dmy x,

9 First Prev Next Last Go Back Full cree Close Quit where the secod equality results from (2.8) ad the third oe from stadard itegratio tools, startig with (2.4). Recall that G y is X -valued ad the, for B, ad x X, we have x dg y = G y () (x) = My x (). O the other had, sice T = θ V, we have T g c x = θ V (g c x ) = θ (g V (c x )), where we are appealig to (3.1) for the idetity V (g c x ) = g V (c x ). By the first part of Theorem 3, it is clear that y θ (g V (c x )) = g V (c x ) (t) d µ (t) y, for each y Y. Now, comparig this itegral to the oe computed above for y T g c x, we get g V (c x ) d µ( )y = g dmy x, for all g C (). ice the scalar measures µ( )y, My x are regular (the first oe by Theorem 3 ad the secod by Propositio 2), it results from the classical Riesz represetatio Theorem that My x () = V (c x ) (t) d µ (t) y, which is exactly (3.2). I the sequel, we wat to improve the estimatio formula (3.2), by suppressig its depedace with respect to the fuctioal y. We will reach a improvemet with the help of the secod part of Theorem 3, sice the formulas give there are more tractable i vector itegratio calculus. To achieve this program we must impose a weak compactess assumptio o the operator T. 4. Weakly compact Operators Let T : F be a bouded operator of the Baach space ito the Baach space F ad let B be the closed uit ball of. The operator T is said to be weakly compact if the weak closure of T B is compact i the weak

10 First Prev Next Last Go Back Full cree Close Quit topology of F. If T : C (, X) Y factors as T = θ V, (see sectio 2), the we have the followig iterestig property: Propositio 3. The operator T is weakly compact iff the operator θ is weakly compact. Proof. Assume θ weakly compact. ice B is bouded V B is bouded ad the T B = θ V B has a weakly compact closure, so T is weakly compact. More importat for us is the coverse. Assume T weakly compact. To prove that the same is true for θ, it is sufficiet, by the berlei-šmulia Theorem [3, Theorem V 6.1.], to show that θa is weakly sequetially compact for every bouded set A C(). Let h be a sequece i A, ad let f C (, X) be such that h = V f ; the, citig Propositio 1, for some K > 0 we may choose f so that f K h for all. This shows that f is uiformly bouded. ice T is weakly compact, the berlei-šmulia Theorem just cited, allows the extractio of a subsequece f i of f such that T f i will be weakly coverget. But T f i = θh i, thus the sequece θh cotais a coverget subsequece, provig that θa is weakly sequetially compact. Remark 2. It is proved i [3, VI.4.5], that for every weakly compact θ ad every bouded V, the product θ V is weakly compact. I the precedig Propositio we were able to get the coverse, that is, θ is weakly compact provided that θ V is weakly compact ad V is oto. While Theorem 4 gives the structure of the adjoit family {G y, y Y }, via formula (3.2), we ow state a improvemet of this formula by imposig o the operator T a coditio of weak compactess. Let γ : Y Y deote the caoical isomorphism of Y ito its bidual Y. Theorem 5. Let T : C (, X) Y be a bouded operator ad assume that T is weakly compact ad factors as T = θ V. The there exists a uique coutably additive vector measure µ o B with values i Y, such that the represetig measure G of T has the followig cosolidated form: (4.1) G () (x) = V (c x ) (t) dγµ (t)

11 First Prev Next Last Go Back Full cree Close Quit for all B ad all x X. Proof. From Propositio 4, the operator θ is weakly compact sice T is weakly compact. Therefore, by the secod part of Theorem 3, µ is a true vector measure o B with values i Y. With this i mid, we proceed as i the proof of Theorem 4 to get y T g c x = y θ (g V (c x )) = g V (c x ) (t) d y µ (t), where the secod equality is from ( b ) of Theorem 3. But y T g c x = ( ) G () (x) (y ) = V (c x ) (t) d y µ (t), g c x dg y, thus we coclude that sice g is arbitrary i C() (see the proof of Theorem 4). Let us put α for the right had side of this last formula; we have by Theorem IV.10.8(f), i [3], α = y V (c x ) (t) d µ (t), ad sice the itegral V (c x ) (t) d µ (t) is i Y, we get α = γ( V (c x ) (t) d µ (t)) (y ); ow let us replace the itegral i ( ) by this value, we obtai G () (x) (y )=γ( V (c x ) (t) d µ (t)) (y ), for each y Y, ad cosequetly G () (x)=γ( V (c x ) (t) d µ (t)). But the last trasformed itegral is exactly V (c x ) (t) dγµ (t), by the Theorem just cited. This achieves the proof of (4.1). There is a iterestig class of operators for which formula (4.1) has a stroger meaig, because the itegrals will be of Bocher type. It is the class of uclear operators which we cosider i the followig sectio.

12 First Prev Next Last Go Back Full cree Close Quit 5. Nuclear Operators Defiitio 4. Let, F be Baach spaces. We say that a bouded liear operator T : F, from ito F, is uclear if there exist sequeces (x ) i ad (y ) i F such that x y < ad such that T (x) = x (x) y for all x X. The followig Theorem gives a itegral represetatio for a uclear operator θ : C () Y : Theorem 6. (i) very uclear operator is compact ad thus weakly compact. (ii) A bouded liear operator θ : C() Y is uclear if ad oly if its represetig measure µ is of bouded variatio ad has a Bocher itegrable derivative g with respect to its variatio v(µ, ), that is µ () = g (s) v(µ, ds). (Recall the variatio of a measure i (2.5).) For the proof see referece [1, p. 173]. We ow tur to uclear operators T : C (, X) Y which have the product form T = θ V. We first give the lik with the uclear property of the compoet θ. Theorem 7. (a) Assume that θ is uclear. The there are sequeces (µ ) C (, X), (y ) Y such that µ y < ad T f = µ (f) y for all f C (, X), so T is uclear. Moreover we have V f = 0 = µ (f) = 0, for all. (b) Assume that the operator T = θ V is uclear ad write T as: T f = µ (f) y, where f C(, X), (µ ) C(, X), (y ) Y ad µ y <.

13 First Prev Next Last Go Back Full cree Close Quit (N ) If the coditio is satisfied the the operator θ is uclear. T f = 0 = µ (f) = 0, for all. Proof. (a) Assume that θ is uclear ad let us write θ as θh = θ (h)y, where (θ ) C(), (y ) Y, h C() ad θ y <. If f C(, X) the V f = h C() ad T f = θh = θ V f y = µ (f)y, where we defie the bouded liear operator µ o C(, X) by µ (f) = θ V f. ice we have θ y <, it follows that µ y < ad T is uclear. O the other had it is clear that: V f = 0 = µ (f) = 0, for all. (b) The coditio imposed to the µ ad T reads Ker T Ker µ. The Ker V Ker µ ad by Theorem 1, ectio 2, with Y = R, for each there exists a bouded operator θ : C() R such that µ (f) = θ V f for all f C(, X). Let h C() ad f C(, X) be such that V f = h; the T f = θh = µ (f)y, but µ (f) = θ V f = θ h. Thus θh = θ (h)y. ice µ y < it follows that θ y < ad θ is uclear. Theorem 8. Let T : C(, X) Y be a uclear operator such that T = θ V. Assume that for all f C(, X), T f = µ (f)y, where (µ ) C(, X), (y ) Y ad µ y <. If coditio (N ) is satisfied for the µ ad T the the represetig measure G of T is a Bocher itegral with respect to a bouded scalar measure. Proof. By Theorem 6 (i) T is weakly compact ad so we have by (4.1): ( ) G()(x) = V (c x )(t)dγµ(t),

14 First Prev Next Last Go Back Full cree Close Quit where µ is the represetig measureof θ. From the coditio imposed o T, we deduce that θ is uclear (Theorem 7) (b) ad the µ() = g(s) v(µ, ds), for a v(µ, ds)-bocher itegrable fuctio g : Y, (Theorem 6 (ii)). Applyig the bouded operator γ to the precedig equality gives γµ() = γg(s) v(µ, ds). O the other had, by a simple argumet of itegratio theory, we have u(s) dγµ(s) = u(s)γg(s) v(µ, ds), for every bouded scalar measurable fuctio u o. Therefore, takig u(s) = V (c x )(s) i formula ( ), we get (5.1) G()(x) = V (c x )(s) γg(s) v(µ, ds) which is the coclusio of the Theorem. 6. xamples We give ow a example of a bouded operator V : C(, X) C(), that meets coditio (2.1) ad the we factorize uder coditio (2.2) a bouded operator T : C(, X) Y. I this cotext we will perform explicitly the computatios made i all of ectios 2 5. Let z be a fixed fuctioal i the cojugate space X of X. The cosider the operator W z : C(, X) C(), give by (W z f)(s) = z (f(s)), f C(, X), s. It is a simple fact that W z is bouded ad that W z = z. Moreover we have: Lemma 1. The operators W z are oto for all z 0.

15 First Prev Next Last Go Back Full cree Close Quit Proof. Let α X be fixed such that z α (α) 0. Let h C() ad let us put f(s) = h(s) z (α), s. The it is clear that f C(, X) ad we have thus (W z f)(s) = z (f(s)) = h(s)z α ( z (α) ) = h(s), W z f = h. It is oteworthy that, i geeral the vector f give above is ot uique. Cosider ow a bouded operator T : C(, X) Y ; to factorize T through W z, with a bouded θ : C() Y, we must assume coditio (2.2) of Theorem 1. I this case, for each g C(), T has the costat value θg o the fiber Wz 1 (g) of C(, X). As a simple example of this situatio take X = R ad z (y 1, y 2,..., y ) = y 1 + y y. The (2.2) reads: f = (f 1, f 2,..., f ) C(, R ), f 1 + f f 0 = T f = 0, ad we have T f = θ(f 1 + f f ), for all f C(, R ). Note also that W z satisfies coditio (3.1). Now, if we wat to compute the represetig measure G of T, all what we have to do, i view of (3.2), (4.1), ad (5.1), is to compute the fuctio V (c x ) for V = W z. This is a trivial matter sice c x is a costat fuctio with value x o : V (c x )(s) = (W z c x )(s) = z (c x (s)) = z (x). Thus formulas (3.2), (4.1), (5.1), become respectively Propositio 4. Let T ad W z be as above ad such that T = θ.w z where θ is bouded. The we have: (a) G y () = (µ() y ) z, for all B ad y Y, that is the X -valued measure G y is geerated by the uique fuctioal z X. (b) If T is weakly compact the G() = (γµ()) z, for all B. (c) If T is uclear the G() = ( γg(s) v(µ, ds)) z, for all B.

16 First Prev Next Last Go Back Full cree Close Quit Now we give a example of a uclear operator which satisfies coditio (N ) of Theorem 7(b). To this ed, let us recall that if Y is fiite dimesioal the every liear operator T : C(, X) Y is said to be degeerate. Propositio 5. If T : C(, X) Y is a bouded degeerate operator, the T is uclear ad satisfies coditio (N ) of Theorem 7(b). Proof. By the [4, Theorem ], a bouded degeerate operator T : C(, X) Y has a represetatio of the form T (x) = µ k (x) y k where {y k, 1 k } ad {µ k, 1 k } are sets of liearly idepedet 1 elemets i Y ad C(, X), respectively. Therefore T is uclear ad by the represetatio above it satisfies coditio (N ) of Theorem 7(b). If dim Y =, the questio arises whether there exist uclear operators T : C(, X) Y which satisfy coditio (N ) of Theorem 7(b). I this cotext, Propositio 5 allows the followig cojecture: Cojecture 1. If Y is a separable Hilbert space, the every uclear operator T : C(, X) Y satisfies coditio (N ). 7. Remark I this work we attempted to give some iformatio about the represetig measure G, which had occured i the cotext of the itegral represetatio (2.7). We obtaied results for the class of factorizable Baach valued operators o C(, X). Let us poit out that similar results had bee obtaied i [5, 5] for aother special class of operators, ad we may summarize as follows. Cosider a bouded operator T : C(, X) X which satisfies the followig coditio: for x, y X, f, g C(, X), if x f = y g, the x T f = y T g. The there exists a uique bouded scalar regular measure o, B such that T f = f dµ for all f C(, X); that is the operator T is a Bocher itegral o the fuctio space C(, X), (ee [5, 5] for more details). Now, accordig to the itegral form (2.7), the operator T has a represetig vector measure G with values i the Baach space

17 L(X, X ). A compariso made by the author i [5, 5], betwee the measures G ad µ, allowed the followig rather precise relatio o the structure of the measure G : (7.1) B G() = µ() γ where γ is the caoical isomorphism of X ito X. Ackowledgemet. I would like to thak the referee for valuable suggestios leadig to the fial versio of the paper. 1. Diestel J. ad Uhl J. J., Vector Measures, Mathematical urveys umber 15, AM, Diculeau N., Vector Measures, Pergamo Press Duford N. ad chwartz J., Liear Operators, Wiley Classics ditio Published Hille. ad Phillips R.., Foctioal Aalysis ad emigroups, Colloq. Publ. AM Meziai L., Itegral represetatio for a class of vector valued operators, Proc. Amer. Math. oc. 130(7) (2002), L. Meziai, Departmet of Mathematics, Faculty of ciece, Kig Abdulaziz Uiversity, P.O Box Jeddah, 21589, audi Arabia, meziailakhdar@hotmail.com. First Prev Next Last Go Back Full cree Close Quit