Internat. J. Math. & Math. Scl.
Vol. 9 No. 4 (1986) 653-658
653
P-REPRESENTABLE OPERATORS IN BANACH SPACES
ROSHDI KHALIL
Department of Mathematics
The University of Michigan
Ann Arbor, Michigan, 4I09, U.S.A.
(Received November 7, 1985)
ABSTRACT.
Let
E and F be Banach spaces. An operator T L(E,F) is called
p-representable if there exists a finite measure u on the unit ball, B(E*), of
Lq(,F), +
and a function g
such that
-=
Tx
E*
x,x*> g( x*) d x*)
B(E*)
for all x
The object of this paper is to investigate the class of all
E
p-representable operators. In particular, it is shown that p-representable operators
form a Banach ideal which is stable under injective tensor product. A characterization via factorization through LP-spaces is given.
KEY WORDS AND PHRASES. Representable Operator, Banach Space, Stable Ideal Operators.
19 AMS SUBJECT CLASSIFICATION CODE. 47BI0.
I.
NT RODU CT ON.
Let L(E,F) be the space of all bounded linear operators from E into F
and B(E*) the unit ball of E*, the dual of E
The completion of the injective
tensor product of E and F is denoted by E
F
Integral operators in L(E,F)
were first defined by Grothendieck, 12], as those operators which can be identified
with elements in
(E
F)*.
These operators turn to have a nice integral representa14], for statements and proofs of such representations.
We refer to Jarchow,
Later on, Persson and Pietsch, [5]
defined p-integral operators in L(E,F) as
those operators T" E
F such that Tx /
<x,x*> dg-(x*), for all x C E*
tion.
where
G
is a vector measure on
B(E*)
B(E*) with values in
lifB(E,) q(x*)dG(x*)ll i(x*)IPd)l/P<(JB(E*)
F
and
for some finite measure
on
B(E*)
The representing vector measure for T
B(E*)
Further, if G is of bounded variation and F
doesn’t have the RadonoNikodym property, then T need not be a kernel integral operator.
The object of this paper is to study operators which are in some sense kernel
ingegral operators. Such operators is a sub-class of Pietsch p-integral operators.
Throughout this paper, if E is a Banach space, then E* is the dual of E and
on
and all continuous functions
need not be of bounded variation.
B(E)
the closed unit ball of
function of
K
If
(,u)
E
If
K
is a set then
is a measure space, then
1K
is the characteristic
LP(.c,,E)
is the space of
R. KHALIL
654
..
p
If
with values in E, for
all p-Bochner integrable functions defr,e on
es3ecia!]y bounded functions on L with values
L’(’,,E) is the space of
p
+
Most of
ir, E.
always denote the conjugate of p
q
The real
P q
refer
[I].
We
Uhl
and
Diestel
and
our terminology and notations are from Pletsch [6]
to these texts for any notion cited but not defined in this paper.
2.
Rp(E,F).
DEFINITION 2.1.
L(E,F) is called p-representable operator if
defined on the Borel sets of B(E*) and a function
x,x* g(x*)d(x*)
and Tx
ilg(x*)ll qd
An operator T
there exists a finite measure
g’B(E*) --F such that
iBfE,,,
,
B(E*)
x E.
It follows from the definition that every p-representable operator is Pietsch-pbe the set of all p-repreintegral operator, but not the converse. Let
sentable operators from E into F.
for all
Rp(E,F)
Rp(E,F)
LEMMA 2.2.
Let
PROOF.
I
+
2"
Rp(E,F)
TI,T 2
Ti(x)
Set
is a vector space.
.
such that
x,x*
gi(x*)di(x*).
B(*)
Then
Consequently,
Further, since
fi d
d;
hi(K) #(K) for all Borel sets K on B(E*), it follows that 0 <_. fi(x*) <__ a.e.
p <-, and
1,2. Let (x*)
gl(x*)fl(x*) + g2(x/)f2(x*). Since
0
fi(x*)
for all x
Lq(B(E*),’,F). Further
we have
This ends the proof.
E.
Rp(E,F), we
llT!lo(p)
For T e
(TI+T2)(XT=
define
inf
{(Illg(x*)l!qd(x*)
I/q
where the infimum is taken over all g and u for which T(x)
!I_ (p) is a norm on
It is not difficult toshowthat
x E
liTi
ilTli
.....
Rp(E
PROOF.
F)
For T
LEMMA 2 3
Let Tx
(
<x,x*>(x*)du,
B(E*)
Rp(E
.
o(p)
<x,x*>g(x*)d(x*) for some
B(E*)
|
<x,x*>g(x*)du(x*),
F B(E*)
and
lg(x *)I !qd(x*)
such that
Definition 2.1. Choose g and
O. Then, using Holder’s inequality"
for a given small
i’T’:o(p)
IITI!
]ITI] (p)
LEMMA 2.Zl.
PROOF.
some
g
E
Let
+c.
Since
Every element T
Tx
_< llTllo(p)_ +
(!, B(E*) ilg(x*)]Iqd(x *) 111q
]ITxI’
Hence
g as in the
I/q
-x,x*
B(E*)
Lq(B(E *),u, F). Choose
+
is arbitrary
Rp(E,F)
the result follows
is an approximable operator in
L(E,F).
g(x*)d..(x*), for some finite measure on B(E*) and
and
g
such that
P-REPRESENTABLE OPERATORS IN BANACH SPACES
Let
gn
gn (x*)lq du(x*)
!g(x*)
IB(E*)
0. Define
such that
Tn(X*)=E,)<x,x*> gn(X*)
lit
is a finite rank operator, and
T
mable operators, Pietsch 16]
Tn
and
Lq(B(E*),., F)
be a sequence of simple functions in
655
O.
(p)
is approximable.
d;:(x*).
Then each
Then by definition of approxiThis ends the proof.
Rp(E,F),
I(p)
THEOREM 2.5. Let H, E,: F and G be Banach spaces, and T
and
B L(H,E). Then ATB
IIBIII:T
Rp(H,G)
PROOF.
measure
u
<x’x*> g(x*)d(x*) for all
fB(E*)
some g Lq(B(E*),u,F). Then
Let Tx
on
B(E*)
and
ATx
(
E and some finite
(
=,
x,x*> Ag(x*)du(x*)
JB(E*)
qdu(x*)
B(E*) flAg(x*) llqdu(x*) _< I!aI, !BiE*) llg(x*)ll
and
x
L(F,G)
A
Hence AT
(
Rp(E
G)
IIATII(p) <- llallllTlio(p)"
To show TB Rp(H,F),
in
Lq(B(E*), F)
and
and
let gn be a sequence of simple functions converging to
T n be the associated operators in R p (E F). So
x,x* gn(x*)du(x*)
TnX
B(I*)
With no loss of generality we assume
into F via:
Gn(K)
g
I!BI
I.
Define the vector measures
G
on
B(H*)
(y*)dG(y*)
B’(E*) K
(B*x*)Sn(X*)du(x*)
B(*) K
Clearly, G n is a countably additive vector measure of bounded variation.
we define the measure \, on B(H*) via
v(K)
B(*)
Further, if
1K(B*x*)d(x*),
then, using Holder’s inequality:
llGn(K)!
BIE*)
.
lign(X*)l!
q d;(x*)
l’q
.[,CK)
I/p
Since the range of G n is finite dimensional, it has the Radon-Nikodym
Gn
F) such that dG n Snd,,.
(B(H*)
property and consequently *here exists S n e
Lq(B(H*)
,F).
S
tha
check
Further, it is easy to
n
An application of the Hahn-Banach theorem, we get:
Hepce
’n BV
By,x*
a
(x*)d...(x*)
B(*)
y,y*.
B(h*)
Sn(Y*)d..(y* ).
B(E*), the sequence (. By,.>g n) is Cauchy in
Lq(B(E *) F). Consequently the sequence(-y,. -S n) is Cauchy in Lq(B(H*), ,,,F). Let
y,. S be the limit of (. y,. -Sn) in Lq(B(H*),,.,F). It is not difficult to see
y,y*. S(y*)d (y*).
that T [I converges in the operator norm to the operator Jy
Since ti function
n
By,*
is pounded on
656
R. KHAL IL
However T n B
TB e
TB
Rp(H,F).
In the operaLor
,ITB
Further
!’T
DEFINITION 2.6.
L(LP(.-.,),F)
(-,)
Let
TBy
hence
I’.
=.f
<y,y*. S(y*)d(y*),
and
B(H*)
This enos 1;he proof.
Bil
(p))
(,,p
Theorem 2.5 states nat
T e
norm.
a normed operator ideal,
is
[6]
F a Banach space.
be a measure space and
is called B-vecLor Integral operator if there exists
An operator
Lq(,p,F)
g
such nat
/ f(t)g(t)d(t)
Tf
for all
LP(,).
f
if the function
g
is only Peztis q-integrable and the integral defining
T
is known to be called veczor integral operator
Now using Theorem 2.5 we can prove-
the Pettis integral, then
THEOREM 2.7.
E,F
Let
T
be Banach spaces and
L(E,F).
Tf
is
[I]
The following are
equivalent"
T e
(i)
Rp(E,F)
(ii) There exists operators T e L(E,LP(2,)) and T 2 e L(LP(,),F)
measure space (a,) such that T 2 is B-vector integral operator and T
PROOF.
(ii).
(i)
Let T e
Tx
Rp(E,F)
on
B(E*)
T2T 1.
and
g(x*)d(x*)
x,x*
for some finite measure
for some
and
Lq(B(E*),,F).
g
Define
LP(B(E*))
T "E
(TlX)(X*)
<x,x*
>,
and
T2
LP(B(E*), )
T2(f)
Then
T2
---->
F
HIE*) f(x*)g(x*)d
is a B-vector integral operator and
TI(E,LP(,))
(ii)
(i). Let T T2T I,
operator in L(LP(.,), F). Then T 2
T2T
Rp(E,F).
Let Ip(E,F)
IFFlli(p)
IFFIli(p)
T
T2T I.
and
Rp(LP(,),
F).
T2
_
is a B-vector integral
Using Theorem 2.6,
This ends the proof.
be the space of Pietsch p-integral operators from E into F, and
be the p-integral norm for T e
Clearly
and
<
for all T
This, together with the fact that
is complete,
Ip(E,F).
Rp(E,F).
IITllo(p)
[5]
THEOREM 2.8.
Rp(E,F)
Ip(E,F)
Ip(E,F)
one can prove"
(Rp(E,F),
llo(p))
is a Banach space.
F has the Radon Nikodym property, then RI(E,F)
Corollary 5 in [i], we see that RI(C(c),F)
II(E,F)
is the class of nuclear operators from E into F.
If
5p(E,F)
II(E,F),
NI(C(p,),F),
Further if
is the class of p-summing operators from E
follows from the Grothendieck-Pietsch represenation theorem [6], that
and by using
where
NI(E,F)
into
F,
then it
Rp(E,F) IIp(E,F)
P-REPRESENTABLE OPERATORS IN BANACH SPACES
3.
657
P,p.
IDEAL PROPERTIES OF
We let R denote the operator ideal of a’ p-representable operators.
P
lowing notions are taken from Pietsch [5] and Holu, 13].
T
(i) An operator ideal
J(E,F) if and only if
J
E and F,
J(,F**), where K F is the natural embedding of F
is called regular if for all Banach spaces
KFT
F**.
Into
(ii) J is called closed if the closure of
all Banach spaces E and F.
J
(iii)
is called injective if whenever
E
for all Banach spaces
F.
and
JF
Here
(B(F*)).
J
(iv)
T
T
The fol-
J(E,F)
JF T
L(E,F)
in
J(E,=(B(F*))),
(
is
J(E,F) for
then
is the natural embedding of
F
T
(E,F)
into
is called stable with respect to the injective tensor product if
T ( T 2 J(E
E 2, F
F2), for all Banach spaces EI,E2,FI,F 2.
THEOREM 3.1. R is regular.
P
PROOF. Let E and F be any Banach spaces and let
Rp(E,F*), for
and g as in
<x,x*> g(x*)d(x*) for some
L(E,F). Then
J(Ei,Fi),
then
KFT
KFTX B(E*)
Definition 2.1.
Now
g(x*)
(
KF(F)
for all
onto operator, the function g(x*)
( Lq(B(E *),, F).
Further
Tx
B([*)
Rp(E,F).
Hence T
B(E*).
Since
Kl(g(x*))
is
x*
KF(F) is an isometric
KF’F
well defined measurable and
<x,x*>](x*)d(x*).
This ends the proof.
In a similar way one can proveTHEOREM 3.2.
R
THEOREM 3.3.
R
PROOF.
Let T
P
P
is injective
is stable.
Rp(Ei,Fi),
1,2
/
x,x*>
gl(X*)d;l(X*)
x,x*
g2(x*)dJ2(x*),
TlX
./"
T2x B(E)
and
ui and gi be the associated measures and functions as in Definition 2.1.
with
1,2, is the completion of the injective tensor product of E
E C$ F i,
(
Ill, then T
T 2 L(E 0 F I, E 2 F2). Further"
,y,y,> g2(Y*)du2(y*).
/
x,x*> gl(X*)dUl(X*)
(T ( T2)(x y)
where
B(E)
B(E)
Let
E2)*.
B(E)
-
B(E)
B(E),
B(E)}
{x* 8 y*" x*
in
y* (
)
Since the map
B Eo*,. the projective tensor product of
E 2, is continuous, [7], it follows that the map
(x*,y*) x* 0 y* is continuous. This induces an isometric into
K be the w*-closure of
(E
E with
If
F i,
B(E),
B(E)
E E
ET
B(E)
R. KHALIL
658
operator
"
C(B(E)
C(K)
there exists a measure
K
on
B(E))
f(z*)d(z*)
K
(
and
@ T
2
z
Consequently,
.2)(x*,y*).
E2)*
E2)*
B(E),
T2)(z)
(T @
T
o-
E2)*----.
B(E),
for all
f
K. Further define
=-0 on B(E @
by putting
F @ F 2 via g(x* @ y*)
gl(x*) @ g2(y *) if x
g(z*) 0 otherwise. Then it is not difficult to see that
B(E @
to
g’B(E @
y*
f o ,(x*,y*)d(
B(E) B(E)
Extend
,(f)
defined by
such that
z,z*>g(z*)d(z*)
B(E @
E2)*
E @ E 2. Since g Lq(B(E @ E2)*,p, F 0
Rp(E @ E 2. F @ F2). This ends the proof.
A negative result for R
P
F2),
it follows that
is the following-
THEOREM 3.4. R is not closed.
P
PROOF: Assume R is closed. Since the ideal of finite rank operator is conP
one has the ideal of approximable operators is contained in
tained in
By
Lema 2.4, one gets R P the ideal of approximable operators. Theorem 2.8, together
and
with the open mapping theorem we get that Ii
I are equivalent on
This is a contradiction. Hence
is not closed.
Rp,
Rp.
ll(p)
Rp
ACKNOWLEDGEMENT. The author would llke to thank Professor Ramanujan and Dr. Defant
for stimulating discussions. This work was done while the author was a visiting
professor at the University of Michigan. The author would also llke to thank the
Department of Mathematics at the University of Michigan for their warm hospitality.
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I.
2.
3.
4.
5.
6.
7.
DIESTEL, J. and UHL, J.J. Vector Measures. _M.athematical Surveys, 15.
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GROTHENDIECK, A. Produits Tensoriels Topologiques et Espaces Nucleaires, Mem.
Amer. Math. Soc. No. 16, 1955.
HOLUB, J.R. Tensor Product Mappings, Math. Ann. 188 (1970) 1-12.
JARCHOW, H. Locally Convex Spaces. Teubner Stuttgart, 1981.
PERSSON, A. and PIETSCH, A. P-Nucleate und p-lntegrable Abbildungen in BanachrIumen, Studia Math. 33 (1969) 19-62.
PIETSCH, A. .Operator Ideal. North Holland Pub. Comp. 1980.
SCHAEFER, H. Topololical Vector Spaces. New York, Macmillan Co., 1966.
Rp.