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Interpolation of compact operators by Goulaouic procedure
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 âuja,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) ââº 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.