# Interpolation of compact operators by Goulaouic procedure by benbenzhou

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Interpolation of compact operators by Goulaouic procedure

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```									C.-H. Chu and B. lochum
STUDIA MATHEMATICA 97 (t) (1990)
64
iran
This is because weakly null sequences in B lift to weakly null sequences inâ¨A. Indeed, let q: A. -Â»B be the quotient map and (bâ) weakly null in B withâ¨q (oj = bâ and (flj bounded in A. Let (uâ) be a countable approximate unit forâ¨kerq and put cn = (1 âuja,r Then q(cn) = bn and (câ) is weakly null in A. Forâ¨the latter, let p be the support projection in A** for kerr/, so that p{ 1â¨strongly in A*'*, which implies p(1 â uâ)aâ->0 strongly. Hence, for feA*, we
un) ââº 0
have f (câ) = f (pcâ) +/ ((1 -p) câ) = f(p (1 - uj aâ) +f ((1 -p) aâ) 0. Now we
Interpolation of compact operators by Goulaouic procedure
have:
by
Theorem, A separable C*-algebra A has the Dunford Pettis property if andâ¨only if A* has this property.
If A has the property, then using the lemma and the proof of Theorem 7,â¨A is type I. Moreover, A has only finite-dimensional irreducible representationsâ¨for otherwise K (12) shows up in a quotient of A. Hence A** is type I finite (cf.â¨Theorem 1 in Hamana's paper).
FERNANDO CO BOS (Madrid)
Abstract. We show that the classical Lions-Pcetre compactness theorems for Banach spacesâ¨(which are the main tools for proving all known compactness results in interpolation theory) fail inâ¨the locally convex case. We also prove a positive result assuming compactness of the operator inâ¨both sides.
1. Setting of the problem. Motivated by certain problems in the theory ofâ¨partial differential equations, Goulaouic studied in [6] and [7] a procedure forâ¨extending any interpolation functor for Banach couples to more generalâ¨couples of locally convex spaces. Let us briefly review this procedure.
A (Hausdorff) locally convex space E is said to be the strict projective limitâ¨of the family of Banach spaces if the following conditions are satisfied:
1)	e â pi tejEf
2)	E is equipped with the projective limit topology.
3)	For each isL E is dense in
4)	The family (E;)ieJ is directed, i.e. given any finite subset J <= I, there existsâ¨Ice I such that for all jsJ the embedding Ek c, E} is continuous.
We then write E = Lim = E,.
References â
[1]	C. A. Akemann, P. G, Doods and J. L. B. Gamlen, Weak compactness in the dual space ofâ¨a C*-algebra, J. Funct. Anal. 10 (1972), 446-450.
[2]	K. Andrews, Dunford-Pettis sets in the space of Bochner integrablefunctions. Math. Ann. 241â¨(1979), 35-41.	"
[3]	J. Arazy, Linear topological classification of matroid C*-algebras, Math. Scand. 52 (1983),â¨89-111.
[4]	J. Bourgain, New Classes of Lt-Spaces, Lecture Notes in Math. 889, Springer, Berlin 1981.
[5]	A. Connes, Classification of injective factors, Ann. of Math, 104 (1976), 73-115.
â [6] J- Diestei, A survey of results related to the Dimford-Pettis property, Contemp. Math.â¨2 (1980), 15-60.
[7]	I. Dobrakov, On representation of linear operators on C0(T, X), Czechoslovak Math. J. 21â¨(1971), 13-30.
[8]	N. Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math.â¨Soc. 47 (1940), 323-392.
[9]	A. Grothendieck, Sur les applications lineaires faiblement compactes d'espaces du tvpeâ¨C(K), Canad, J. Math. 5 (1953), 129-173.
[10]	G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, 1979,
[11]	S. Sakai, C*-Algebms and W*-Algebras, Springer, Berlin 1971.
[12]	M. Takesaki, Theory of Operator Algebras I, Springer, Berlin 1979.
[13]	S. K. J, Tsui, Decompositions of linear maps, Trans. Amer. Math. Soc. 230 (1977), 87-112.
iej
Let now (A0, At) be a (compatible) couple of locally convex spacesâ¨(meaning that they are continuously embedded in a Hausdorff topologicalâ¨vector space). We say that (.A0, Ax) is the strict projective limit of the familyâ¨(A0iJ, /4ij)(!,/)Brxi of Banach couples provided that the following conditions
hold:
1)	A0 = Lim AQj, Ax Lim Auh
isi
2)	All spaces A0ih AXlJ are continuously embedded in a common Hausdorffâ¨topological vector space ,rJ,
3)	For each (iJ)elxJ, A(jnAi 'is dense in	(the norm inâ¨Ao,tr\Atj being max {||a|U0>(, HU,,})-
DEPARTMENT OP MATHEMATICSâ¨UNIVERSITY OF CALIFORNIAâ¨Irvine, California 92717, U.S.A.
GOLDSMITHS- COLLEGEâ¨London SEW 6NW, U.K.
UNIVERSITY DK PROVENCE andâ¨C.N.R.S., CENTRE DE PHYSIQUE THfiORIQIJE
JsJ
Luminy Cass 907, F-I32KH Marseille Cede* 9, France
Received June 9, 1989â¨(addendum received September 18, 1989)
(2572)
1985 Mathematics Subject Classification: 46M35, 46A45,
5 - Studia Mathematics 97,1
icm
Interpolation of compact operators
67
F. Cobos
66
Lions-Peetre Lemma. Let 0 < 0 < 1, 1< q ^ co, let {A0, A J be a Banach
couple and let B be a Banach space. Assume that T is a linear operator.
(i)	If T: A0-*B is compact and T: A 1~*B is continuous, then T:â¨(A0, is compact.
(ii)	If T: B^A0 is compact and T. B-> A, is continuous, then T:â¨B-r(A0, Ai)o,q is compact.
(In fact, Lions and Peetre showed that this is true for any interpolationâ¨method of exponent 0 and not only for the real method.)
The aim of this note is to show that the Lions-Peetre Lemma fails for theâ¨Goulaouic procedure. We also prove a positive result of Hayakawa type.
If this is the case, we write
(Aq, Aj) â Lim(A0i, Al j).
i.j
Any interpolation functor for Banach couples F can be extended toâ¨projective limit couples by defining the interpolated space as the projective limitâ¨of the family {F(A0J, AUj)}{Lj)<,IXJ:
F(A0,Al)^UmF{AOJ,AUj).
ij
As an example, consider the Scbwarz classical space fyLp of all infinitelyâ¨differentiable complex functions / defined in R", with D" f e Lp for everyâ¨multi-index a. Then we have
2. The counterexample. First let us recall the definition of the echelon spaceâ¨of order p > 1.
Let (am,J be an infinite real matrix such that
0 < a
] consists of all sequences c = (Â£.,) of scalars such that for
0! â¢^Lpjo.p &ip.
Here 1 ^ p0, pj < oo, 0 < 6 < 1, 1/p = {l â 0)!pQ + 0/p1 and ( , )(liJJ denotes theâ¨real interpolation method (see. [11] and [12] for details 'on this method).
In general, if (A0, Ax) â Lim{y4o ;, A1J) then the topology of [A0, is
' '
defined by the family of norms
(*)
m, n = 1,2,...
< am +1
,111
m,ri
The space lp[aâ¨every weN
m,n
co
vJÂ®=K\\ipiaâ,n) = (l(am,n\Ur)llP<K>,
n~ 1
and its topology is defined by the sequence of norms vm. See [9], [13], and [1]â¨for details on these spaces.
In order to see that the Lions-Peetre Lemma (i) fails for the Goulaouicâ¨procedure, take
h.M = [f {t 9-Ky(f> a))f dtjt]x^
a
where KitJ- is the Peetre K-functional associated to the couple (v40tI-, AtJ9 i,e.
Kij(t, a) â inf+r0 = 00+0^ a0eA0J, axeAxJ.
Besides the Riesz type formula (*), Goulaouic derived in [6] and [7] manyâ¨other properties of this interpolation procedure, but there is no result thereâ¨(nor in the subsequent literature) on the stability of compact operators for thisâ¨procedure. Accordingly, we study this problem here.
The behaviour of compactness under interpolation is a very naturalâ¨question for applications of interpolation theory to other branches of analysisâ¨and thus has received attention from the beginning of abstract interpolationâ¨theory. The first result in this direction was obtained in 1960 by M. A.â¨Krasnosel'skii [10] for the case of /.^-spaces. Other contributions are due toâ¨Lions-Peetre [11] and Hayakawa [8], among others. But in fact, the questionâ¨whether Krasnosel'skii's result holds true in abstract interpolation does notâ¨have a complete answer yet.
Quite recently new approaches to some classical results have beenâ¨developed in [2]-[5], also yielding new compactness theorems. Surprisingly,â¨the following result established in 1964 by Lions and Peetre [11] plays a mainâ¨role in the proofs of all (new and old) compactness theorems.
(m/(m + 1))",
and let T be the identity operator 77; = Â£. Note that
m, n = 1,2,...,
Â®m,n a2m + 1 ,r
m, n = 1, 2, ...
Thus the restrictions T: l2 l2 [flm,â] and T: l2 [a,2â,â] -Â» l2 [aâ¨tinuous. In addition,
< L
] are con-
m,n
CO
yi Â®m,n/Â®m + t.n "d , m 1,2,...
11= I
Hence the Frechet space l2 [Â«â,,â] is nuclear (see, e.g., [13], Chap. II, 8 3,4(1))â¨and consequently any bounded subset of l2 [<vJ >s relatively compact. Thisâ¨implies that T: i2 -+ l2 [Â«m â] is compact.
Nevertheless, T: (l2, L [a,i,â]j1/2,2->/2[um.â] is not compact. Indeed, theâ¨couple (l2, l2 [Â«!â]) is the strict projective limit of the sequence of Banachâ¨couples (l2, I2{alj)
('2' '2 [flM,n])l/2,2 = him (!2, l2 (Â«m,n))l/2,2 = Lim l2 (flfn.n) = h J â
""	meiN
] is not compact.
Therefore, using [12], Thm. 1.18.5, we obtain
weN â
meN
me.N
And clearly the identity map of l2 [a
r
Interpolation of compact operators
F. Cobos
69
ran
68
Next we show that the Lions-Peetre Lemma (ii) also fails ill the locallyâ¨convex case. Take now
References
[1]	F. Cobos, A new class of perfect Frechet spaces, Math. Nackr. 120 (1985), 203-216.
[2]	F. Cobos, D, E. Edmunds and A. J. B. Potter, Real interpolation and compact linearâ¨operators, J. Funct. Anal. 88 (1990), 351-365.
[3]	F. Cobos and D. L. Fernandez, On interpolation of compact operators, Ark. Mat.â¨27 (1989), 211 217.
flm.nl = l+(n+ l)2m,
m,nâ 1,2,..
â  >
(fli,n/Â«2,n) = 0, the embedding from l3(a2,â)
and let again = C. Since limâ¨into l2(aUn) is compact. Hence T: /2[am,â]-/2 is a compact operator.
?l ~* CO
[4] F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn-Gagliardo Junctors,
Moreover,
Israel J. Math. 68 (1989), 220-240.
al,n < 2a4
so that T: l2 [am,â] -+ i2 [Â«&,â] is continuous. But anew
T: 12 [fl,â,n]	1-2	1/2,2 = '2 [am,n]
m, n = 1,2,...,
[5]	M. Cwikel, Real and complex interpolation and extrapolation of compact operators, preprint.
[6]	C. Goulaouic, Prolong ements defoncteurs d'interpohtion et applications, Ann. Inst. Fourier
(Grenoble) 18 (1968), 1 98.
Interpolation entre ties expaces localement comexes definis d I'uide de semi-groupes; cat.- desâ¨expaces de Gevrey, ibid. 19 (1970), 269-278.
[8]	K. Hayakawa, Interpolation by real method preserves compactness of operators, J. Math.â¨Soc. Japan 21 (1969), 189-199.	'
[9]	G. Kothc, Topological Vector Spaces I, Springer, Berlin 1969.
[10]	M. A. Krasnosel'skii, On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960), 229-231.
[11]	J. L. Lions and J. Peetre, Sw une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci.
[7]
is not compact.
3. A positive result. We close this note by proving that under the hypothesisâ¨of compactness in both sides, the interpolated operator is also compact.
Theorem. Let the couples (T0, T{) and (B0, be the strict projective limitsâ¨of the families of Banach couples (A0ii, A1j\iyj]eIxj and (B0,sÂ» ^i,z)(j,ncsx4>â¨respectively. Assume that T is a linear operator such that T: Ak->Bk compactlyâ¨for k = 0, 1. Then if 0 < 0 < 1 and 1 < q ^ oo, T: {Aa, Ax)(liq^>-{B(1, B^g^ isâ¨also compact.
Publ. Math. 19 (1964), 5-68.
[12] H, Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland,â¨Amsterdam 1978,
[13] M. Valdivia, Topics in Locally Convex Spaces, North-Holland Math. Stud. 67, Amsterdam
1982.
II '.Aoj-*B0 and T:
Proof. Find is I and jeJ such that T: (Aâ¨{An II	are compact. Put
DEPART AMEN TO DE MATEMATICASâ¨FACULTAD DE CIENCIASâ¨UNIVERSIDAD AUT6NOMA DE MADRIDâ¨28049 Madrid, Spain
0'
U = {ae(A0, Ax){)zq: rizj(a) ^ 1}.
We are going lo show that T(U) is precompact in (B0,
Given any seS, zeZ and e > 0, by the density of T0 in d0 Aj in AhJ andâ¨A0nAl in do,;"', A1J; we can extend T to an operator f such thatâ¨~	and T: Alij^-BltZ are compact, and T\Aa+M = T. Then, using
[2], Thm. 3.1 (the extended version of Hayakawa's result), we see that
T: (A0it,	Bl z)eq
(2574)
Received June 12, 1989
T: A
-+B
OJ
is compact. It follows that
T: ((A0, AL)3iq, riJ-y(B0,st Bla)e<qâ¨is also compact. Hence, there exists a finite set {al5 ..., aj c U such that
T(U)<= U (T(ak) + {be(B0,s, BxJo,f rf,Ab)^lj).
k = l
Finally, if beT(U)-T(ak) then be{Ba, Bj)9A and therefore
T{U) <= (J {T(ak) + {be(B0, BL)e,q: r*Jb) ^ 1}).
k~ 1
This completes the proof.

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