24 Decomposing Operators

24.

Decomposing Operators

The famous Dunford-Pcttis-Phillips theorem states that every weakly compact operator X from Ll(Q,p) into any Banach space E can be represented in the foriii Xf =

J x(0)f ( w ) 440)

for all

f E L,(Q,y )'

0

where x is a p-measurable E-valued function defined on the probability space (Q, p). A Banach space E is said to have the Radon-Nikodym property if this result remains true for all operators. I n this chapter we deal with thc ideal 1 ' ) of so-called Radon-Nikodym operators is that of decomposing and give its main properties. The dual ideal 0 = '1)'"' operators. We also introduce the ideal Q, of p-decomposing operators with 0 < p 5 oc. It turns out that Q, = p y and 8, = Q y for 1 < p < 03. Using the concepts described above we can improve some multiplication theorems for absolutely summing and integral operators. In particular, Grothendieck's formula ?lBo 3 = 92 is established.

24.1.

Measurable Vector Functions

24.1.1. In what follows ( Q , y ) always denotes any probability space. By an Ecalued function x we mean a map from SZ into an arbitrary Banach space E. 24.1.2. An E-valued function x is said t o be p-sinaple if n

Xih,(Cu),

x(w) = I

...,xn E E and ...,Qn of 9.

where q, subsets Q1,

h,, ...,h, are characteristic functions of y-measurable

24.1.3. An E-valued function 2 is called y-measurable if there exists a sequence of p-simplo E-vtllud functiocs a, such that x(to) = l k xn(o)ahnost everywhere. n

24.1.4. Let us mention the trivial Lemma. For every p-measurable E-valued function y-measurable, rts well.

d:

the scalar funclion

I@(.)\\i s

24.1.5. Now an important characterization of p-measurable vector functions is stated. For proofs mo refer t o [DIU, p. 421, [DUN, p. 1491, and [IOX, p. 721. Theorem. An E-valued function x i s ,u-nreasurable i f and only if the f o l l o w i q ccnditions are satisfied: iL (1)There exists a separable subspace A4 such that x(w) E iW almost everywhere. (2) All scalar functions (x(.), u) with a E E' are p-measurable.

336

Part 5. Applications

41.1.6. Lemma. Let x be cc fi-measurable E-valued function. Suppose that for all a E UE, we have I(z(o),a ) / p almost everywhere. Then 112(0~5 )!1p wlinost everywhere.

Proof. First we observe that alniost all values of x belong to some separablc subspace M of E. Let ( x n ) be a countable dense subset of M . Furthermore, pick a, 5 UE#with (xn,ajL)= Ilz,J/. Then = 1,2,

llzll = sup (i(x, a,)! :

...)

for all x E X.

Finally, we can find a p-null set A' such that I(z(o), a,)l

2 p for all w E Q \ A'

and

12

= 1, 2,

...

This yields Ilz(m)ll (= e almost everywhere. 2 1.1.7. I n what follows we do not distinguish between p-measurable E-valued functions which coincide alniost everywhere. The collection of these equivalence classes is denoted by L,(E, Q, ,u). Let

Ilzllo:= inf ( p )= 0: p(w E 12: Ilx(w)ll > e) 5 e j . Using standard techniques we get the T h e o r e m . Lo(E,0, p ) is a complete metric Zineur r p c c with the P-norm l~.llo.

21.1.8. If z is any p-simple E-valued function of the form 7k

a(o)= 2 x i k i ( o ) , 1

the11

1z ( w ) dp(c0) 2 Zip(Qi) :=

?I

1

is well-defined: cf. 24.1.2.

24.1.9. An E-valued function x is called p-integrable if there exists a sequence of p-simple E-valued functions x, such that lim IIz - zn/io = 0 and II

hm.nn

j

//Z,,~(O>)

n

-

x,(o)ll d p ( o ) = 0 .

Then the Bochner integral is well-defined by

J z ( w ) d,u(w) := lim J x,,(to) d,u(to). n

'I

D

94.1.10. By [DIU, p. 451 we have the Lemma. An E-valued functim t is p-inteyrcrble if md only if it is p-tueu8urable and dP(0) < 60-

J Il=(c.)l 12

24. Decomposing Operators

24.2.

337

Radon-Nikodym Operators

24.2.1. An operator X E f?(L,(Q,p), E) is called r@ht decomposable if there exists a ,u-measurable E-valued function x such that

X f = J x(w)f ( w )++J)

for all f E L,(Q, p ) .

R

Then x is said to be the kernel of X . 24.2.2. The following lemma can be derived froin 24.1.6.

Lemma. The kernel x of every right decomposable operator X E 2(Ll(12,p), E ) i s unique almost everywhere. Horeover,

llXIl

= ess-sup ( l ~ x ( w ) w ~ lE: Q ) .

24.2.3. We next state the famous Dunford-Pettis theorem a proof of which can be found in [BOU,, chap. VI, p. 461, [DIE, p. 2251, and [DUN, p. 5031. Theorem. Let E be a B a M space such that E' is separable. Then every operator X E s(L,(Q,p ) , E') is right decomposable. 36.2.4. We are now able to check the deep Theorem. Let E be a reflexive Banach p e . Then every operator 3 E f?(Ll(Q,p), E') is right decomposable.

Proof. We know from 3.2.9 that L,(Q, p ) has the Dunford-Pettis property. So every operator X E f$(L,(O, p ) , E ) is completely continuous. Observe that the embedding map I from L , ( Q , p ) into L,(Q,p) is weakly compact. Thus, by 3.1.3, the product X I must be compact. Consequently 1M := lM(X1)is a separable subspace of E . Since L,(Q, p) is dense in L,(Q, p), we also have M ( X ) = M . So X acts from Ll(Q, p ) into M = MI', and we can apply 24.2.3.

24.2.5. As an immediate consequence of 2.4.3 and of the preceding result we get the Dunford-Pettis-Phillips Theorem. Every operator X E !Xl(L,(Q,p), E ) is right decomposable. Remark. A direct proof is given in [ION, p. 891. See also [DIU, p. 751.

f-1.2.6. Let S 2 ( E , F ) . Then S is called a Radon-Nikodynz operntor if S X is right decomposable whenever X E B(L,(Q,p), E ) . The class of all Radon-Nikodym operators is denoted by g. 24.2.1. Theorem. 9 .is a closed operator d e a l .

Proof. It can be easily seen that m

of operators S, E g ( E , P)with are kernels yn such that

s , X f = J y,(w) n

22 Pietsch. Operator

9 is an operator ideal. We now take a sequence

llSnli< OQ. Given X E 2(L1(D,p), E),then there

1

* f(w) dp(w)

for all

f E L,W, P I .

338

Part 5. Applications

By 24.2.2 we have I]yn(w)]] 5 l]SnXilalmost everywhere. Since ca

m

2' IlgAw)!I 5 Z IIfL!' !!XiI, 1 1 by setting Qi,

Y(0) :=

z

Yn(c.)

1

we get a p-measurable P-valued function which is defined almost everywhere. So we m

have found a kernel y of the operator SX, where S:=ZS,,. 1 S E g ( E , F).Hence the operator ideal 9 is closed.

This proves that

22.2.8. Theorem. The operator ideal 'I) is injective. Proof. Let S E %(E,P).Then, given X E B(Ll(s2,p), E ) , there exists a kernel y such that

S X = ~ J g ( w ) /(oj) d,u(w) for all f E L,(SZ,p ) . R

-

Put B := M ( S X ) . Applying 24.1.6 to the FIN-valued function Q;y y(w) E d l ( 8 X ) almost everywhere. This proves the injectivity of 9.

we see that

24.2.9. A Banach space belonging to Y := Space ('I)) is said to have the RadoaNikodym property. Remark. The thcory of these Banach spaces is presented in [DIE] and [DIU].

242.10. Proposit ion. The B a m c h space 1, possesses the Radon-Nikodym property. Proof. The assertion follows from 24.2.3, since 1, is a separable dual Banach apace.

24.2.11. Proposition. The Banuch spaces Ll[O, 11 and co fail to have the R&ANikodym property. IZ

Proof. Let I be the countable set of all indices (k,n ) with k Porin the intervals

= 1, 2,

...

= 1,

...,2" and

and denote the corresponding characteristic fiinctions by hkn. Then the equation Xf:= ( f , hkn)defines an operator S E B(L,[O, 11, co(I)).Let us assume that L,[O,11 E Y or co(I)E Y. Then X is right decomposable, and we can find a co(l)-valuedliernci X. Obviously z(t)= (hkR(t))almost everywhere. On the other hand, we always have (hkn(t)) 6 since card ((k,n):hkn(t)= 1)= No for all t E (0, 11. This contradicticjn implies L,[O, 11 ( Y and c o ( l ) Y. Clearly, %(I) and co are isoiuorphic.

24.2.12. Proposition. !ll3 c 'I). Proof. The inclusion follows from 24.2.5. Moreover, we have lI E Y \ W.

24. Decomposing Operators

339

24.2.13. Proposition. QJ c 8. Proof. Suppose that some S E QJ(E, F ) is not unconditionally summing. Then by 1.7.3 there exists X E B(co, E) for which SX is an injection. Now the injectivity of QJ implies that the identity map of c,, belongs to B. This is a contradiction.

24.2.14. Proposition. The operator deals ?Band QJ are incomparable. Proof. Let X be t,he operator constructed in 24.2.11. Then X E B \ QJ.On the for ll all illfinite dimensional reflexive Banach spaces E . other hand, we have Ix E B \ ?

24.3.

Vector Measures

be any a-algebra given on a set 9. A map m from $' 3 into a Banach 24.3.1. Let space E is called a vector measure if

for every sequence of disjoint subsets B,, B,,

..- E $' 3.

34.3.2. The variation po of a vector measure m is defined by

The supremum is taken over all finite families of disjoint subsets B,, belonging to %.

...,B, of B

24.3.3. The proof of the following lemma can be found in [DIN, pp. 32-35]. Lemma. The variation of every vector meamre is a smlar measure which is, however, 7zeces8arilyfinite.

not

24.3.4. We now consider a vector measure m and a probability p, both defined on the same a-algebra 8.Then wz is called p-continuous if p ( B ) = 0 implies m ( B ) = o for B E 8.The E-valued measure m is said to be p-differentiable if there exists a p-integrP+bleE-valued function x such that

m ( ~=)J z(w)d,u(o) for all B

c B.

B

Then x is called the Radon-Nikodym dericative of m. 24.3.5. Let m he any E-valued measure. Then the P-valued ineasurc Stti, where 8 E B(E,F ) , is defined by ( S m ) (B):= X[rn(B)]for B 8.

21.3.6. Every p-differentiable E-valued measure m is p-continuous and has finite variation. The fact that the converse stakment is false gwve rise to define RadonNikodyin operators; cf. [DIU, p.501. Theorem. An operator X E 2 ( E , F ) is a a d o n - N i k o d y m operator i f and only i f it ?naps every p-continuous E-valued memure m with finite variation h t o a p-differcntiable F-valued measure Xm.

.

22*

340

Part 5. Applications

Proof. Given X E B(Ll(Q,p), E ) , then m ( B ):= X(hB) for B E % defines a y-continuous E-valued measure m with finite variation. Let us now assume that S possesses the property mentioned above. Then there exists a derivative y of am. This means that = Jy(w)

SX(ILB)

~ B ( w dy(w) )

for all B E 8.

0

Consequently we get

SXf =

J Y(0)f ( w ) 4 4 w )

D

for all p-simple functions f. By continuity the same formula holds for all f E L,(Q, p). This proves that 8 E g ( E , F ) . To check the converse implication we consider a p-continuous E-valued measure m with finite variation po. Then there exists an operator X E B(Ll(12,h),E ) uniquely defined by the condition X(hg) = m ( B ) for all B E %. If 8 E g ( E , F ) , then we can find a p-measurable F-valued kernel yosuch t,hat

8x1= J ? l o b ) fb)dpo(w)

for all f E LdQ, Po)

-

Q

Since the p-continuity is transmitted from m to po, according to the classical RadonNikodym theorem, there is a y-integrable scalar function g 2 0 such that ,tAo(~)=

J g(w) d p ( w )

for all B E B .

I3

Consequently, by putting y := gy,, we get Xm(B) = S X ( ~ B ) = J yo(@) +,(to) B

=

1y(o) dp(cr>).

B

s o y is the derivative of Sm. This proves that every Radon-Nikodym operator has the required property.

24.4.

Decomposing Operators

24.4.1. An operator A 6 B(E, L,(S, p)) is called left decomposable if there exists a y-measurable E-valued function a such that

Ax

= (2,

a(.)) for all x E E .

Then a is said to be the kernel of A . 24.4.2. An operator S 6 B(E, F ) is called decomposing if BS is left decomposable whenever B E f?(F,L&2, p)). The class of all decomposing operators is denoted by a.

24. Decomposing Operators

341

TZ = 9"'. P r o o f . Let S E Q(E,P).If Y E B(L,(sZ, p ) , F'), then Y'K,S is left decomposable.

24.4.3. Theorem.

For x E E and f E &(Q, p ) we have (2,S'Yf) = !Y'Kp!h, f

J (x, ~ ( w )f ) (

)=

~d )p ( ~ ) ,

Q

where a is the corrcsponding kernel. The above equation yields

S ' Y ~= J a ( o ) f ( o ) d,p(w) for a11 f E L,(Q. p). Q

This proves that x't'

g(F',El). Hence D

gaual.

Conversely, let X E (I)*""(E,F ) . If B E g ( P ,L,(O, p)),then S'B'KL1is left decomposable. For x E E and f f L,(D, p ) we have

I(%,N o ) )

{B~X f ) ,= (x,S'B'K,,f) =

f(0))

~Aw),

s1

where a is the corresponding kernel. The above equation yields BSx = (2,a(.))for all x E E. This proves t8hatS E Q(E, P).Hence gd*' Q. 24.4.4. As a counterpart of the preceding result we formulate the

P r o p o s i t i o n . Qdual c 9. ( ~ ) d u a l ) d u n5 l Oreg = g. In order to = (Ddual implies Qdual Proof. Clearly show that Qdw1 + 9 we consider the related space ideals. By 24.2.10 we have I , E Y. On the other hand, the assumption I , E ad"'implies 1'; E Y. Since 1'; and LJO, 11" are isomorphic [SED, p. 4801, the injectivity of Y yields Ll[O, 11 E Y, which is a contradiction by 24.2.11.

24.5.

p-Decomposing Operators

243.1. Let' 0 < p m. An operator A f B(E,L J 9 , p ) ) is called left decomposable if there exists a p-measurable El-valued function a such that

A x = (2,a(.)) for all x E E . Then a is said to be the kernel of A . 24.5.2. Lemma. The kernel i.s unique almost eoeryuhere.

of

every left decomposable operutor A E B(E;L,(Q, p ) )

Proof. Obviously it is enough to show that. m y kernel a of the zero operator vanishes almost everywhere. First we observe that almost all values of a belong to some separable subspace M of E'. Let (a,) be a countable dense subset of M and choose x, E U, such that 2 i(xn, a,)[ 2 Ilu,ll. Then llallO

:= sup (I(x,, .}!:

n = 1, 2,

...I

342

Part 5. Applications

defines an equivalent norm on M . More precisely, we have 2 lla/lo2 /lull 2 jjaljOfor all a E M . Take a p-null set N such that (zn,~ ( w ) = ) 0 for all

OJ

E s;! \ N

and n

=

1,2, ...

Hence l l ~ ( a ) )= //~ 0 almost everywhere. This proves the assertion. 24.5.3. Let 0 < p 5 00. An operator S E B(E,F ) i3 called p-decomposing if BS is left. decomposable whenever B f E(P,Lp(s;!, p)).

The class of all p-deconiposing operators is denoted by

Qp.

Remark. Obviously, we have Q = Q,. 24.5.4. We formulate without proof the trivial

Theorem. Q p is an operator ideal. 24.5.5. Using the method of 24.2.8 we get the

Theorem. The operator ideal Q p is surjective. 24.5.6. Theorem. If 1

p 5 00, then the operutor ideal Q, i s regular.

Proof. The regularity of 0, follows from 4.5.6 and 24.4.3. Let S E Q'dig(E,P)and 1 < p < 00. Then, given B c B(F, Lp(Q,p)), by 1.5.5 therr exists BzE B(F", L,(D, p ) ) such that B = B7KF.Consequently BS = BT(KFS) is left decomposable and therefore S E Qp(E,F ) . This proves that Q, = QFg. To treat the case p = 1 we need the fact that there is a canonical surjection Q from L,(D, p)" onto L,(B, p ) which can be defined by the help of Lebesgue's decomposition; cf. [BOU,, chap. V, p. 611. Then the left decomposibility of BS folloas from BS = (QB")(K,S). Remark. It seeins to be unknowii whether or not the operator ideal 0,is also regular for 0 < p < 1. 24.5.7. Theorem.

Qp

s FpdpU8l.

Proof. Let S E Qp(E,F),Then for every 6 E E(F, Lp[O,11) there exists a unique Lebesgue-measurable E'-valued kernel a such that BSrr = (x,a(.))for all x E E. Clearly B +u defines a linear map K froiii B(F, Lp[O,11) into Lo(E',[0, 11). We noTs consider a sequence of operators B,, B,, E E(F, L,[O, 11) converging to some operator B f E(F, Lp[O,11). Furthermore, let us assume that the sequence of corresponding kernels a,, C I ~ ., .. tends t o some vector function a E Lo@',[0, 13). Then

...

= Lp-lim(2,un(.)) = Lp-limBnSx = BSx. Lo-lim (r,an(.)) n

(I

11

On the other hand, by hypothesis we have Lo-lini(2,an(.))= (x,a ( . ) ) . n

Hence u is the kernel of BS. This proves that the map K is closed arid therefore continuous. Choose 6 > 0 such that \\I311 5 6 iniplies I/allo5 1/2. We claim that

24. Decomposing Operators

343

for all finite families of functionals b,, ...,b, E F‘. In order to verify (*) we inay assume that wp(bi)= S and I[S‘bi[l> 0. Put l i

II~‘bilP

+ + IIfJ’bmll”

:=

IlS’blllp and form the intervals

..*

If h,, ...,h, are the corresponding characteristic functions, t,hen bi @ hi

B := defines an operator B

B(P,L,[0, 11) with ijB[j= 6. Since

m

1

is the kernel of BS, it follows that /\ullo5 lj2. On the other hand, we have that i:n(t)ll= lp(8’bi)for all t E (0,11. So lp(8’bi)> 1/3 would iinply I!alio> 1/2, which is a contradiction. Therefore (*) holds, and t,his means that S’ E !J&,(F’,El). Consequently

a, 5 qy.

31.5.8. To prove the main result of this section we need thc following Lemma. Let 1 p < 00 and A E p y ( E , L,(Q, ,LA)).Then there emkts f i L,(Q, p ) such that \Ax1 5 f

for all

2

E

U,.

I n other terms, A maps the closed unit ball U , into an order bounded subwt of Lp(!2)p). Proof. Take any finite p-measurable decomposition (Ql,

..., Q m ) of B such that

p(Qi) > 0, and denote the family of corresponding characteristic functions by (hl, .,hm).Defineu,, u, E Lp(Q,p) and cl, v,,, E L,.(Q, y ) by ui := ,u(Qi)-’%i and vi := , U ( B ~ ) - ~ ’Then %~.

...,

..

...,

m

L := 1z’j 0ui 1

is an operator in LJQ,,u);of. 19.3.5. It follows from wP(vi) 5 1 that lp(A’vi)5 Ppd‘(A). SO w

fL

:=

2 [IA’viIiui 1

fulfils llfLllp 5 P;mi(A).-Since

344

Part 5. Applications

..

Consequently max (JLAzl:,I, ., ILAs,l) 5 fJ for all finite families of elements x,, ..., x,,E U,. This yields llniax (\LAX,!, ..., ~LAx,l)ll,2 P y ( A ) . Observe that, LAz tends to Ax, if (9,; ...,9,) ranges over all finite p-measurable decompositions of 9 ;cf. 19.3.5. So, passing to the limit, we have

..

Since the family of all functions max (IAxll, ., IAx,]) is directed upwards, according to [BOU,, chap. IV, p. 1371, there exists a supreinum f E Lp(9,p). Clearly f has the required property.

We are now able to establish the fundamental Theorem. If 1 < p

< M, tkcn. Q p

=

,y

and

Qiml= Qp.

Proof. Let S E Q,(E, 3') and X E O(E', Lp(9,p)). According to 17.3.11 without loss of generality we may suppose that B' is reflexive. By the preceding lemma there exists a positive function f E L p ( 9 ,p ) such that JXS'b/5 f for all zi E UFt.Hence Yob := XS'b/f defines an operator Y o B(F', L,(O, p)). It follows froin IF E 9 that IF, E Q. Thus we can find a p-measurable F-valued kernel yo such that P,b = (yo(,),b) for all b 6 li". Hence XS'b = (g(.),b) for all b E F', where y := /yo. Therefore S' is p-decomposing. This proves that Q, & Qy. As proved in 24.5.7 we also have Q p & So we get

vy.

!ppE Q y l & ( v y ) d u a l s vp and dual Q,dual c =(Qp

dual

c

=DpSPyl.

24.5.9. As a counterpart to the preceding theorem we have the Proposition. Q,

+ Py'.

Proof. The canonical map J , from C[O, 13 into L,[O, 11 is 1-integral. Hence J , E 3p1 5 gp$".We now assume that J1E Q1. Then J1has a Lebesgue-measurable C[O, 11'-valued kernel j,. Form the polynomials pn(t):= tn for n = 0, 1, ..., and choose a Lebesguc-null set N such Ohat

pt2(t) = (p?87j1(t)) for au. E

Lo, 1'

\N*

Sccording to Weierstrass's approximation theorem the above equat.ion holds for all functions f E C[O, 13. This means that j,(t) = 8(t) for all t E [0, 13 \ N .

However, there does not, exist any separable subspace M of C[O, 11' contailling almost all Dirac measures 8(t). This contradiction proves that J , 6 Q,. 24.5.10. Finally, we state the trivial Proposition. I/ p1 5 p2,then Qpl E Q,,. Remark. Using a famous theorem of E. M. NIKISHIN [i] it can be shown that the deal Q, with 0 < p < 1 does not depend on the parameter p .

24. Decomposing Operators

24.6.

345

Multiplication Theorems

24.6.1. I n order t o prove the basic result of this section we need a preliminary Lemma. Let (0, p ) be any probability space and 1 5 r < 00. Then for every left decomposable operator iZ E f?(E,L,(.Q, p ) ) the product 1,A is r-nuclear such that S,(I,A) 5 114 .11. Here I , denotes the embedding map from L&2, p ) into &(0, ,u). Proof. First we treat the special case in which the kernel a of A has a countable a(o)= aj}.Without loss of generality we may assume image (ai).P u t Qi := ( w E 0: that p(Sj)> 0. Let f i := ,u(Oi)-l/%i,where hi denotes the charact,eristio function of R,. Ther! 'x1

1,A

=

2,' ,u(R~)~!' C C ~S J fi. 1

It follows from

lr(p(Ot)l'r)= 1 , wm(ai)=r llAll, and wrt(fi)= 1 , that I,A is an r-nuclear operator with N,(I,A) 5 1 1 8 1 1 . We now come t o the general case. Since almost all values of the kernel a belong to some separable subspace H of E', we can find B sequence of p-measurable Elvalued functions a, possessing a countable image such that ess-sup (Ila(co) - an(co)i/: w 0) l/n. P u t A,z := (x,an(.))for all

5

E E. Then we have A = /l.ll-lim A,. Moreover, I

N,(I,A, - IrAn)5 llAm- A,li implies that (I,A,) is a n X,-Cauchy sequence. Since I,A is the only possible limit, we get I,A E a t , ( E ,L,(Q, p)) and

N,(I,A) = lim Nt,(I,A,) 5 lim liAnlI = IIAl]. n

n

We are now able to establish the fundamental Theorem. Let 1 2 r P r ,

< 00. Then

I,] 0 [a.Il-lil S

[a,,K P g .

P r o o f . Let T E Q(E,F ) and S E &(F, G ) . Given E > 0, by 19.2.6 we can find a factorization K,S = ZI,B such that IiZIl /IB// (1 c) Ir(8). Since BT is left decomposable, the preceding lemma yields KGST = Z ( I , B T ) E %,(E, G") and

+

Nr(KGfiF) 5 11211 Br(IrBT) 5 llZllllsrll 2 (1

+ &)

lITll.

This proves that S1' E ?J2Fg(E,(2) and lKf""(ST)5 I,(&')l!2'l.

24.6.2. As a consequence we get the famous Grothendieck Theorem.

[m, 11.111 o [Z, I] = [n,N].

Proof. By 2.4.3 every S E m ( F , G ) factors through some reflexive Banacli , and 2 E %(Go,G ) space Go. More precisely, given E > 0, we can find SoE f ? ( P Go) such that S = 28, and liZI[llSoll5 (1 E ) IISII. Now 24.2.12 yields Sh E gad((&, P'). So, by 24.4.3, we hare 8; E Q((2&3"). Let T E 3 ( E , F). Then T' E Z(F', E') and

+

:%46

Part 5. Applications

therefore T‘S;

%(GA, E’) as well as N(1”SA) 5 I(T’) IiSblI. Csing the reflexivity

of Go we get ST = Z(S,,T) E R(E, G) and

N(ST) 5

llzll N(SoT) i IIZlI IISoIl I(T) 2 (1

This proves that [!& 8.6.4.

11.11]

o [3,I]

[a,N].

+

E)

IiSII I(T).

The converse inclusion is evident by

R e m a r k . Let us niention that a n operator S E B(E,F ) belongs to 9 if and only if S-Y is nuclear for all strongly integral operators X E B(Eo,E ) ; cf. [DIU, p.1751.

24.6.3. We next state soine k i d of Grothendieck’s formula for r-integral operators. P r o p o s i t i o n . Let 1 4

T

< m. Then

[R,11.111 0 P r , 1 7 1 = [%,> 5 7 1 . Proof. Suppose that T E 3,(E, F ) and S E R(F, G). According t o 19.2.6 there exists a factorization KFT : YI,A such that !lYll llAij 5 (1 t F ) I,(!Z’). We have S7 %(F”, G). Since L,(Q,,u)has the approximation property, the operator S7Y is e v m approxiinable. So 19.1.10 yields ST = (S”Y) (IJ) E %,(E, G) a n d

S,(LSY)5

llsq ilAlj 5 (1 + 8) I’SIj I,(T).

r ,

I his proves that

[R,ll.lll 0 P,>I,] E [anKI. The converse inclnsion follows from 19.2.2. R e m a r k . Clearly

[m,ii.ii]

o [3,,I,]

4[a,,S,] for t

< r < 00.

24.6.4. We are now ready to improve 20.2.4. T h e o r e m . Let l / r

+ l/s = 1,‘p5 1 cxnd 1 <

T,

s < m. Then

i):oof. By 17.3.11 every operator T E P,(E, 3’)admits A factorization T = Y T , such that T , < ‘&(E, P o ) , I? < 2 ( F o , F ) , and liYil Ps(To)5 (1 -1 t ) P,(T), where F , IS >: rafltxive Banach space. If X :3,(B’, G), then 24.6.1 iinplies K,SY E %,(Po,G“) a d X,(K,SY) 5 I,(S) IIY~~. Thrrefoie KGST == (K,SY) To E ?Rp(E,G”) and

Np(K,ST) 5 S,(K,SY) P,(?‘o) 5 (1

-c1

p:ores

i’z-:tig

+ F) I,(#) Ps(7’).

p).

to t!ie iiijective Iinlls froin (3) we get (1).

Let 7’ S s ( E , P ) and S I3 Y 3 ( F ,G). Then XT E 3FJ(F”, G) and 1;’ (&IT) = VJ(S). By 13.2.6 there is a factorization K,T = Y1,A with IlYll IlAll 5 (1 F ) Is(T). Since L6(l2,p ) is reflexivc, the operator Y is decomposing. So the injective version

+

24. Decomposing Operators

347

of 24.6.1 (theorem), namely

[Sr,Irlinj 0 [Q, 11.111 G [!It,. hTrlin', yields STY %F(Ls(s2,p), G ) and S:"j(S7Y)5 p ( S )llYIl. By 19.2.16 we can find a factorization S"Y = ZSo, where doE 3y(Ls(s2, p), Go),Z E A(Go,G),and

llZll I:"j(S0)5 (1 + e ) I:"j(S)IjY11. Altogether we get the diagrmi :

~,(-Q,

r ) T & ( Q ,r ) T . a o

Then S0IJ E S J E , Go) and Ip(#,,18A)5 I:"J(SO) llA/l. Finally, it follows from the preceding proposit'ion that ST = Z(SoIsA)E ?JIp(E,G) and

N,(ST) 5 IlZll &(SOLA) 5 (1

5 (1

+

C)Z

+ ).

1 3 4 IlYll

ll4

I:"?(#) 18(T).

This proves (2).

24.6.6. As a special case of the prtwding result we formulate the Theorem. [?&,PJ2 E [!It,N].

24.7.

Notes

Therc is an extensive list of papers dealing with representations of operators in function J. PETTIS[l], I. AT. GELFAXD spaces. We only mention the classical a o r k of N. DUNFORD/B. 111, and R. S. PHILLPS [i]. A full presentation and further references may be found in [DUN, pp. 489-5111. The theory of Radon-Nikodym operators has been developed by W.TJNDE [3] and 0. J. REINOV[i]. For further informattions the reader is referred t o i~ survoy paper of J. DIEsTa/J. J. Urn, [l] and the monographs [DIE] and [DIU]. Tii? coricept of a p-decompocf. [BAD, exp. I?$ Pitrther contributions sable operator was introduced by A. BAUBIKIAX; d r C due to S. KWAPIER [6], P. SAPHAR133, and L. SCHWAETZ [2], [3]. Th- most striking multiplication theorem of this chapter first appeared in [GRO. chap. 3. p. 1321. Other important results are taken f r o i A. PERSSOX 121. Becoxurnendations for furtlirr reading:

[DIN], [ION], [SEn/r,, annex I], [SElil,. csp. Oj. [SEA&, exp. -1-6.51, [SEi\I,. esp. 11, [swal,

[TAR]. S. BOCHNER [l], J. DIESTEL[i], V. K. K O ~ T K O[if, V [2], W. LINDE[B], 131, J. VON KEU[I], A. PIETSCH [4], [9], J. W. RICE[l], R. ROGGE [ I ] , C. SWARTZ [Z],G. I. TABGONSKI [I], E.THOMAS [I], [2], [3], J . WEIDNANN[l], J. WLOKA[l], [a], T. K. TVom 111, [2], [3].