UNIFORM FACTORIZATION for COMPACT SETS of OPERATORS Introduction the Purpose of This Note Is to Obtain a Factorization Result Fo

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 4, April 1999, Pages 1119–1125 S 0002-9939(99)04619-5

UNIFORM FACTORIZATION FOR COMPACT SETS OF OPERATORS

R. ARON, M. LINDSTROM,¨ W. M. RUESS, AND R. RYAN

(Communicated by Theodore W. Gamelin)

Abstract. We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonn´e Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

Introduction The purpose of this note is to obtain a factorization result for relatively compact subsets of the Banach space of all compact weak*-weak continuous linear maps using a Banach space version of Michael’s Selection Theorem which Bartle and Graves [BG] proved in the early fifties. Precisely, they showed that if X and Y are Banach spaces and u is a continuous linear map from Y onto X, then there exists 1 a continuous map f : X Y such that f(x) u− (x) for all x X. In addition to the Selection Theorem,→ our approach is to∈ use Grothendieck’s∈ characterization of relatively compact sets in the projective tensor product and factorization results of compact operators through a universal Banach space, due to Johnson [J] and Figiel [F]. We further present a second method of proof for our factorization result, based on the Banach-Dieudonné Theorem. From our main theorem we obtain extensions of results of Graves and Ruess [GR]. Their methods are different and, in our opinion, more complicated. We also obtain a new proof of a result of Toma, characterizing polynomials that are weakly uniformly continuous on bounded sets.

1. Preliminaries Generally, our notation and terminology are standard and we refer to the books [Di] and [G]. For the deﬁnition and properties of p-spaces the reader is referred to [LT]. L

Received by the editors July 25, 1997. 1991 Mathematics Subject Classification. Primary 46B07; Secondary 46B28, 46G20, 47A68. Key words and phrases. Banach spaces, compact factorization, tensor products, Michael’s selection theorem, Banach-Dieudonnétheorem. This note was written while the second and the fourth authors were visiting Kent State Uni- versity to which thanks are acknowledged. The research of Mikael Lindström was supported by a grant from the Foundation of Abo˚ Akademi University Research Institute.

c 1999 American Mathematical Society

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Lw (X0,Y)[Kw (X0,Y)] is the space of all [compact] weak*-weak continuous linear∗ operators with∗ the usual operator norm. There is an isometric isomorphism

K(X, Y ) Kw (X00,Y)[W(X, Y ) Lw∗ (X00,Y)] given by T T 00,where K(X, Y )['W(X,∗ Y )] denotes the space of' all [weakly] compact operators7→ from X into Y . The closed unit ball of a Banach space X is denoted BX .Xc0 [Xτ0 ] denotes the dual of X endowed with the topology c(X0,X)[τ(X0,X)] of uniform convergence on the [weakly] compact subsets of X. c [ τ ] denotes a c-[τ-] neighborhood base of zero consisting of closed absolutely convexU U sets (disks). If U [ ], we denote ∈Uc Uτ by X(0U) the completion of the quotient of X0 by the nullspace of U, endowed with the norm defined by the Minkowski-functional of U. If C X is a closed bounded ⊂ disk, we denote by XC the span of C in X, endowed with the norm given by the Minkowski-functional of C. In [F], [J], the authors proved that there is a universal Banach space Z such that every operator T K(X, Y ) can be factored as T = v u, where u K(X, Z)and ∈ ◦ ∈ v K(Z, Y ). In particular, Z canbechosenasZ=( W C W)p,1 p , ∈ ⊂ p ≤ ≤∞ where W runs through the subspaces of C (and where, as usual, p = is the p P ∞ c0-sum). We can show that Z also serves as a universal factorization space for Kw (X0,Y)- operators. One way to see this is by an application of Johnson’s factorization∗ methods to the norm closure in L(X0,Y) of the finite-rank, weak*-weak continuous operators. Another way is the following approach, which makes use of [Ru1, Thm. 1.7 (e)]: if T Kw (X0,Y), there exists U c such that T (U) is relatively compact in Y. (For∈ details,∗ compare (2) – (6) of∈U the method 2 of proof of Theorem ˜ 1 ˜ 1 1 below.) Defining T : X0/U − (0) Y by T (x0 + U − (0)) := T (x0), and extending → to X(0U) by continuity, T can be decomposed in the following way:

id π U T˜ (1.1) X0 X 0 X 0 Y. −→ c −→ ( U ) −→ Factoring T˜ through Z, the conclusion of the above is the following result. (1) For every pair of Banach spaces X, Y, and for every T K (X0,Y) there ∈ w∗ are operators u K (X 0,Z) and v K(Z, Y ) such that T = v u. ∈ w∗ ∈ ◦ If X or Y is an 1-space (resp. an -space), then Randtke [R2] (resp. Dazord [D], cf. also RandtkeL [R1], Johnson [J])L∞ has shown that every operator in K(X, Y ) factors compactly through l1 (resp. c0).

2. The results Given Banach spaces X, Y , let the universal Banach space Z be as above. Ac- cording to (1), the continuous, bilinear map

τ : K (X0,Z) K(Z, Y ) K (X0,Y),τ(u, v)=v u, w∗ × → w∗ ◦ ˆ is onto. The linearizationτ ˆ : Kw∗ (X0,Z) πK(Z, Y ) Kw∗ (X0,Y)ofτis a continuous linear onto map. Therefore we⊗ can apply the→ Bartle and Graves selection theorem which asserts that there is a continuous map σ : K (X0,Y) w∗ → Kw (X0,Z)ˆπK(Z, Y ) such thatτ ˆ σ = idK (X ,Y ). We remark that the lin- ∗ w∗ 0 earization argument⊗ is necessary if we◦ wish to apply the Bartle and Graves selection theorem. Indeed, C. Fernandez [Fe] has recently shown that there are continuous bilinear surjections τ : X Y Z between Banach spaces X, Y and Z for which there is no one-sided inverse.× →

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Theorem 1. Let X and Y be Banach spaces. For every relatively compact subset H of K (X 0,Y) there exist an operator u K (X 0,Z), a relatively compact subset w∗ ∈ w∗ BT : T H of K(Z) and an operator v K(Z, Y ) such that T = v BT u for {all T H∈. } ∈ ◦ ◦ ∈

Proof. Method 1. By continuity, the set σ(H) is relatively compact in Kw∗ (X0,Z) ˆ K(Z, Y ). Now, by Grothendieck [G, p. 51], there exist null sequences (r )in ⊗π i Kw∗(X0,Z)and(si)inK(Z, Y ) and a relatively compact subset K of l1 such that for each T H we can write σ(T )= ∞ λT r s where λT =(λT) K. ∈ i=1 i i ⊗ i i ∈ Define r : X0 c0(Z)byr(x0)=(ri(x0)). That r Kw (X0,c0(Z)) follows → P ∈ ∗ directly from ri 0. For each T H define AT : c0(Z) l1(Z)byAT(z)= (λTz), z =(zk) k→c (Z). Since ∈ → i i i ∈ 0 ∞ ∞ λT z sup z λT , i i ≤ k ik· i i=1 i i=1 X X we have A L(c (Z),l (Z )). Now consider the continuous map A : l T ∈ 0 1 1 → L(c0(Z),l1(Z)) defined by A(λ)z =(λizi). Since A(K) AT :T H , it follows that the subset A : T H of L(c (Z),l (Z)) is relatively⊃{ compact.∈ } Now we { T ∈ } 0 1 define a compact operator s : l (Z) Y by s(w)= ∞ s (w ), w =(w) l (Z). 1 → i=1 i i i ∈ 1 Compactness of s follows from si 0. Sinceτ ˆ σ = idK (X ,Y ), we conclude k k→ ◦P w∗ 0 that T =ˆτ(σ(T)) = ∞ λT s r and so T = s A r. Finally, we factor r i=1 i i ◦ i ◦ T ◦ and s through Z. Thus, there exist operators u Kw (X0,Z), α K(Z, c0(Z)), P ∈ ∗ ∈ β K(l1(Z),Z)andv K(Z, Y ) such that r = α u and s = v β.Let B∈=βA αfor each T∈ H.Then B :T H is a◦ relatively compact◦ subset T ◦T◦ ∈ { T ∈ } of K(Z)andT=v BT ufor every T H. Method 2: The following◦ ◦ facts will be∈ needed: (2) Given any compact disk C in Y, there exists another such, C1 say, with C C such that the C -topology of Y restricted to C is equal to C. ⊂ 1 1 C1 k·kY | (Simply take C1 = n∞=1(nC +(1/n)BY ). ) (3) c(X 0,X) is the finest locally convex topology on X0 agreeing with c(X0,X) T on all nBX0 ,n N.(Banach-Dieudonné Theorem.) ∈ Now, if H K (X0,Y) is relatively compact, then ⊂ w∗ (4) H(BX0 ) is relatively compact in Y. Since, accordingly, H∗(B ) K a compact disk in X, Y 0 ⊂ 1 (5) H(K1◦) BY . ⊂ ( 1) Let U = ∞ (nB +(1/n)H − (B )); then U by (3) and (5), and n=1 X0 Y ∈Uc (6) H(U) K2 a compact disk in Y. (This follows from (4) and H(U) T⊂ ⊂ nH(BX0 )+(1/n)BY for all n N, and, actually, is a special case of [Ru1, Theorem 1.7 (e)].) ∈ Choose a compact disk K in Y related to K2 according to (2). Then we have: (7) Every sequence (Tn)n H has a subsequence that is uniformly Cauchy over U with respect to the K-topology.⊂ (This follows from operator-norm relative com- ( 1) pactness of H, together with U nBX +(1/n)H − (BY ) for all n N,H(U) ⊂ 0 ∈ ⊂ K2 and K K2 = Y K2.) | k·k | ( 1) ( 1) Now, given T H, define T˜ : X0/U − (0) Y by T˜(x0 + U − (0)) = T (x0), ∈ → K and extend continuously to X(0U). We then have the following uniform factorization for the operators T H : ∈ id π U T˜ idK (2.1) X0 X 0 X 0 Y Y. −→ c −→ ( U ) −→ K −→

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Now, factor both (the compact weak*-weak-continuous map) π id through Z as U ◦ πU id = r u, u Kw (X0,Z),r K(Z, X0 ), and (the compact map) idK as ◦ ◦ ∈ ∗ ∈ (U) id = v s, s K(Y ,Z),v K(Z, Y ), and let B = s T˜ r K(Z). Then, by K ◦ ∈ K ∈ T ◦ ◦ ∈ (7), BT T H is relatively compact in K(Z), and T = v BT u, T H. This completes{ | the∈ proof.} ◦ ◦ ∈

The above method 2 of proof also applies to the case of Lw -operators, except that the factor space Z may depend on X and Y. Speciﬁcally, the∗ following result holds. Proposition 2. Let X and Y be Banach spaces. There exists a reﬂexive Banach

space Z = Z(X, Y ) such that, for every relatively compact subset H of Lw∗ (X 0,Y),

there exist an operator u Lw∗ (X 0,Z), a relatively compact subset BT : T H of W (Z) and an operator∈v W (Z, Y ) such that T = v B u for{ all T H∈. } ∈ ◦ T ◦ ∈ Proof. Given the assumptions of Proposition 2, in method 2 of proof above, state- ments (2) to (7) hold with C and C1 in (2) weakly compact, c(X0,X) in (3) being replaced by τ(X0,X) (the assertion then following from the fact that Xτ0 is a gDF- space, cf. [Ru2]), and, in (4) – (6), “compact” being replaced by “weakly compact”. Altogether, the corresponding reasoning thus leads to the following uniform factorization of the T 0s in H:

id π U T˜ idK (2.2) X0 X 0 X 0 Y Y, −→ τ −→ ( U ) −→ K −→ for some U and some weakly compact disk K Y. At this point, let ∈Uτ ⊂ Z = Z(X, Y ):=( R ) ( R ) , C,U 2 ⊕2 K 2 C,UX XK where U runs through an Xτ0 -neighbourhood base, C runs through the weakly compact disks in X(0U), and K runs through the weakly compact disks in Y, and the corresponding R-spaces are the associated reﬂexive Banach spaces in the Davis-

Figiel-Johnson-Pelczynski factorization [Di] for the range spaces X(0U) and Y, respectively. Applying now the factorizations corresponding to (1.1) and (2.1) to the mappings of (2.2) completes the proof.

Corollary 3. Let X and Y be Banach spaces. For every relatively compact subset H of K(X, Y )[W(X, Y )], there exist an operator u K(X, Z)[W(X, Z)], a relatively compact subset B : T H of K(Z)[W∈(Z)] and an operator { T ∈ } v K(Z, Y )[W(Z, Y )] such that T = v BT u for all T H.(Here,the spaces∈ Z are those of Theorem 1 and Proposition◦ ◦ 2, respectively.)∈ In the compact case, this corollary can be combined with factorization results through nice spaces for compact operators between special spaces. In the following result, we apply Corollary 3 when X is an 1-space or an -space and obtain L L∞ Theorem 2.1 in [GR]. When Y is an 1-space or an -space, a similar result can be stated. L L∞

Corollary 4. Assume that X is an 1-space (resp. an -space ). For every relatively compact subset H of K(X, YL) there exist an operatorL∞ p K(X, l ) (resp. ∈ 1 p K(X, c0)) and a relatively compact subset QT : T H of K(l1,Y) (resp. of K∈(c ,Y)) such that T = Q p for all T H. { ∈ } 0 T ◦ ∈

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Proof. Assume that X is an 1-space (resp. an -space). We apply Corollary 3; L L∞ then, as we have pointed out in the preliminaries, u factors compactly through l1 (resp. c ) such that u = q p.PutQ := v B q and we are done. 0 ◦ T ◦ T ◦ Finally, we show how Corollary 3 yields a new proof of a result of E. Toma [T]. Recall that (nX) denotes the space of continuous n-homogeneous polynomials on X. EachP such polynomial P is associated with a unique element A of the space s(nX)ofn-linear, symmetric mappings on X, satisfying P (x)=A(x, ..., x) for eachL x X.Thespaceofn-homogeneous polynomials that are weakly uni- ∈ n formly continuous on the unit ball of X is denoted P wu( X) and the corre- s n sponding space of symmetric n-linear forms is denoted wu( X). For each n- L s n 1 homogeneous polynomial P there is a linear operator TP : X ( − X), defined n→L by TP (x1)(x2, ..., xn)=A(x1,x2, ..., xn).Pbelongs to P wu( X) if and only if the n operator TP is compact; furthermore, if P Pwu( X), then TP takes its values s n 1 ∈ in the space wu( − X) [AP]. The following corollary is shown in [T]; we offer a different proof,L based on the above results.

Proposition 5. Let X be a Banach space and n a relatively compact subset of s n 1 H the space K(X, wu( − X)). Then there is a compact subset K0 of X 0 such that for L n all T n and all x X, T(x)(x, ..., x) supk K k0(x) . ∈H ∈ | |≤ 0∈ 0 | | Proof. We proceed by induction on n =2,3, ....Let 2be a relatively compact H s subset of the space K(X, X0) of compact linear maps from X to wu(X, C)=X0. By Corollary 2, there are a Banach space Z, a relatively compactL subset L : T { T ∈ of K(X, Z), and an operator w K(Z, X0) such that T = w L for all H2} ∈ ◦ T T 2.Thus,foreachx Xand for each T ,we have ∈H ∈ ∈H∈ T (x)(x) = w L (x)(x) = L (x),wt(x) , | | | ◦ T | |h T i| t regarding x X X00, and so T (x)(x) L (x) w (x) . Now, ∈ ⊂ | |≤|| T || · || || t LT (x) =sup LT (x),z0 =sup x, LT (z0) sup x, k0 || || z B |h i| z B |h i| ≤ k K |h i| 0∈Z0 0∈Z0 0 ∈ 10 t where K0 = L (z ):T ,z B is easily seen to be compact. Furthermore, 1 { T 0 ∈H2 0 ∈ Z0} t t w (x) =sup w (x),z =sup x, w(z) =sup x, k0 , || || z BZ|h i| z BZ|h i| k K|h i| ∈ ∈ 0∈ 20

where K0 = w(B ) is compact. Therefore, if K0 = K0 K0 , then 2 Z 1 ∪ 2 2 T (x)(x) sup k0(x) | |≤k K | | 0∈ 0 for all T and all x X. ∈H2 ∈ Assume now that the result is true for all j

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s n 1 Now w(z): z 1 n 1 is a relatively compact subset of wu( − X), which { || || ≤ }≡H − L s n 2 by [AP] means that n 1 is a relatively compact subset of K(X, wu( − X)). By H − L the induction hypothesis, there is a compact subset K20 X0 such that for all x X n 1⊂ ∈ and all z B , w(z)(x, ..., x) sup k0(x) − . Letting K0 = K0 K0 , it Z k0 K20 1 2 follows that∈ | |≤ ∈ | | ∪ n T (x)(x, ..., x) sup k0(x) | |≤k K | | 0 ∈ 0 for all T and all x X, and the result is proved. ∈Hn ∈ Corollary 6 ([T]). For any n, a continuous n-homogeneous polynomial P belongs n to wu( X) if and only if there is a compact subset K0 of X 0 such that P (x) P n | |≤ supk K k0(x) for all x X. 0∈ 0 | | ∈ n s n 1 Proof. If P wu( X), then TP : X wu( − X) is compact and the result follows by applying∈P the proposition to →LT . Hn ≡{ P} Conversely, suppose there exists a compact subset K0 of X0 such that P (x) n | |≤ supk K k0(x) for all x X.LetJbe the polar of K0 in X and let π be the 0∈ 0 | | ∈ canonical mapping of X onto the Banach space XJ associated with J.Nowthe

dual of XJ is (X0)K0 and it follows that π is compact. Hence π is weakly uniformly continuous on the unit ball of X. But by the assumption P factors through π. Therefore P is weakly uniformly continuous on the unit ball of X.

Added in proof The authors are indebted to K. Floret for pointing out that in Method 1 of the proof of Theorem 1, the Bartle-Graves selection theorem is not needed. In fact, by the lifting property of quotient mappings for compact sets, there is a compact set ¯ L K (X0,Z)ˆ K(Z, Y ) such thatτ ˆ(L)=H. ∈ w∗ ⊗π References

[AP] R. M. Aron, J. B. Prolla, Polynomial approximation of diﬀerentable functions on Banach spaces,J.ReineAngew.Math.313 (1980), 195-216 MR 81c:41078 [BG] R. Bartle, L. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413 MR 13:951i [D] J. Dazord, Factoring operators through c0, Math. Ann. 220 (1976), 105-122 MR 54:8332 [Di] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984 MR 85i:46020 [Fe] C. Fernandez, A counterexample to the Bartle-Graves selection theorem for multilinear maps, Proc. Amer. Math. Soc. 126 (1998), 2687–2690. CMP 98:13 [F] T. Figiel, Factorization of compact operators and applications to the approximation property, Studia Math. 45 (1973), 191-210 MR 49:1070 [G] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires, Amer. Math. Soc. Memoirs 16, Providence, RI, 1955 MR 17:763c [GR] W. Graves, W. Ruess, Representing compact sets of compact operators and of compact range vector measures,Arch.Math.49 (1987), 316-325 MR 89a:46145 [J] W.B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345 MR 44:7318 [LT] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math., vol. 338, Springer, Berlin, 1973 MR 54:3344 [R1] D. Randtke, A structure theorem for Schwartz spaces, Math. Ann. 201 (1973), 171-176 MR 48:4691 [R2] D. Randtke, A compact operator characterization of l1, Math. Ann. 208 (1974), 1-8 MR 49:3507 [Ru1] W.M. Ruess, Compactness and collective compactness in spaces of compact operators,J. Math. Analysis Appl. 84 (1981), 400-417 MR 83h:47032

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[Ru2] W.M. Ruess, [Weakly] Compact operators and DF spaces, Paciﬁc J. Math. 98 (1982), 419- 441 MR 83h:47034 [T] E. Toma, Aplicacoes holomorfas e polinomios τ-continuous, Thesis, Univ. Federal do Rio de Janeiro, 1993

Department of Mathematics, Kent State University, Kent, Ohio 44240 E-mail address: [email protected] Department of Mathematics, Abo˚ Akademi University, FIN-20500 Abo,˚ Finland E-mail address: [email protected] Fachbereich Mathematik, Universitat¨ Essen, D-45117 Essen, Germany E-mail address: [email protected] Department of Mathematics, University College Galway, Galway, Ireland E-mail address: [email protected]

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