REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

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1 REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of C(K) spaces, wth values n an arbtrary Banach space. Ths class has been consdered recently by several authors (see, f.., [3], [8], [9]) n connectons wth problems of factorzaton of polynomals and holomorphc mappngs. We also obtan several characterzatons of the compact dspersed spaces K n terms of polynomals and multlnear forms defned on C(K) Mathematcs Subject Classfcaton: 46E15, 46B Introducton and Notatons Let K be a compact Hausdorff space. C(K) wll be the space of scalar valued contnuous functons on K, Σ wll denote the σ-algebra of the Borel sets of K and B(Σ) wll stand for the space of Σ-measurable functons on K whch are the unform lmt of elements of Σ-smple functons. As t s well nown, the Resz representaton theorem gves a representaton of the operators on C(K) as ntegrals wth respect to Radon measures, and ths has been very frutfully used n the study of the propertes of C(K) spaces. In a seres of papers (see specally [6], [7]), Dobraov developed a theory of polymeasures, functons defned on a product of σ-algebras whch are separately measures, that can be used to obtan a Resz-type representaton theorem for multlnear operators defned on a product of C(K) spaces. Before gong any further, we shall clear out our notaton: If X s a Banach space, X wll denote ts topologcal dual and B X ts closed unt ball. L (E 1..., E ; Y ) wll be the Banach space of all the contnuous -lnear mappngs from E 1 E nto Y, and P( X; Y ) the space of contnuous -homogeneous polynomals from X to Y,.e., the class of mappngs P : X Y of the form P (x) = T (x,..., x), for some T L (X,... X; Y ). When Y = K, we wll omt t. We shall use the conventon. [].. to mean that the -th coordnate s not nvolved. We shall denote the semvaraton of a measure µ by µ and also the semvaraton of a polymeasure γ by γ (for the general theory of polymeasures see [6], or [14]). It seems convenent to recall here that a polymeasure s called regular f t s separately regular and t s called countably addtve f t s separately countably addtve. We wll denote the set of the bounded semvaraton polymeasures defned n Σ 1 Σ wth values n X as bpm(σ 1,..., Σ ; X). rcapm(σ 1,..., Σ ; X) stands for the subset of the regular countably addtve polymeasures and bsvω rcapm(σ 1,..., Σ ; X ) for the subset of bpm(σ 1,..., Σ ; X ) composed of those polymeasures that verfy that for each x X, x γ rcapm(σ 1,..., Σ ; K). Both authors were partally supported by DGICYT grant PB

2 2 FERNANDO BOMBAL AND IGNACIO VILLANUEVA As customary we wll call rca(σ; X) the set of regular countably addtve measures from Σ nto X. Wth these notatons at hand we can state for further references the followng theorem from [4], whch extends and completes prevous results of Pelczyns ([11]) and Dobraov ([7]): Theorem 1.1. ([4]) Let K 1,..., K be compact Hausdorff spaces, let X be a Banach space and let T L (C(K 1 ),..., C(K ); X). Then there s a unque T L (B(Σ 1 ),..., B(Σ ), X ) whch extends T and s ω ω separately contnuous (the ω - topology that we consder n B(Σ ) s the one nduced by the ω -topology of C(K ) ). Besdes, we have 1. T = T. 2. For every (g 1,. [].., g ) B(Σ 1 ). [].. B(Σ ) there s a unque X -valued bounded ω -Radon measure γ g1 on K,...,g [] (.e., a X -valued fntely addtve bounded vector measure on the Borel subsets of K, such that for every x X, x γ g1 s a Radon measure on K,...,g [] ), verfyng g dγ g1, = T (g...,g [] 1,..., g 1, g, g +1,..., g ), g B(Σ ). 3. T s and ω ω sequentally contnuous (.e., f (g n) n N B(Σ ), for = 1,..., and g n ω g, then lm n T (g1 n,..., g n) = T (g 1,..., g ) n the σ(x, X ) topology. Besdes, f we defne γ : B(Σ 1 ) B(Σ ) X as γ(a 1,..., A ) := T (χ A1,... χ A ), then γ s a polymeasure of bounded semvaraton that verfes (a) T = γ. (b) T (f 1,..., f ) = (f 1,..., f )dγ (f C(K )) (c) For every x X, x γ s a regular (scalar) polymeasure and the map x x γ s contnuous for correspondng wea- topologes n X and (C(K 1 ) ˆ ˆ C(K )). Conversely, f γ : Σ 1 Σ X s a polymeasure whch verfes (c), then t has fnte semvaraton and formula (b) defnes a -lnear contnuous operator from C(K 1 ) C(K ) nto X for whch (a) holds. Therefore the correspondence T γ s an sometrc somorphsm between L (C(K 1 ),..., C(K ); X) and the polymeasures n bsv-ω - rcapm(σ 1 Σ ; X ) that verfy condton c). Our am now s to explot both representaton theores, measures and polymeasures, to study the multlnear operators on C(K) spaces. In ths paper we present some results n ths drecton. 2. The Man Results The followng defnton can be found n [6] or n [14] Defnton 2.1. A polymeasure γ : Σ 1 Σ X s sad to be unform n the ı th varable f t s countably addtve and the measures { } γ(a 1,..., A 1,, A +1,... A ) ca(σ ; X) : (A 1,. [].., A ) Σ 1 [] Σ are unformly countably addtve. A polymeasure s sad to be unform f t s unform n every varable.

3 REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 3 It s easy to chec that gven a natural number r, 1 < r < and r ndexes 1 j(1) < j(2) <... < j(r), and gven fxed h j(p) B(Σ j(p) ), p = 1... r, we can construct the multlnear operator T hj(1),...h j(r) : C(K q ) X 1 q q (j(1)...j(r)) defned as T hj(1),...h j(r) (h q(1),..., h q( r) ) := T (h 1,..., h ) whose assocated polymeasure we wll call γ hj(1),...h j(r). Gven a bounded polymeasure γ : Σ 1 Σ X and a fxed number, 1, we can construct n a natural way the measure φ : Σ bpm(σ 1, [], Σ ; X) defned as φ (A ) := γ A. The fact that φ s bounded, ndeed φ = γ, and the followng lemma are easy to chec. Lemma 2.2. Wth the above notaton, a countably addtve polymeasure γ s unform n the ı th varable f and only f φ s countably addtve. The same s true f n ths statement countably addtve s replaced by regular. Let E 1,..., E, X be Banach spaces. Each T L (E 1,..., E ; X) generates n a natural way lnear operators T : E L 1 (E 1,. [].., E ; X), = 1,..., defned as T (x )(x 1,. [].., x ) := T (x 1,..., x ) for each x j E j, j = 1,...,. We wll state now a defnton: Defnton 2.3. A -lnear mappng T L (E 1,..., E ; X) s sad to be regular f every mappng T above defned s wealy compact. When X s the scalar feld, the above defnton was gven n [3]. In general, gven an operator deal U, we can defne the U-regular -lnear mappngs as those such that the correspondng T belong to U for every 1. When U s the deal of compact operators, such mappngs have been consdered n [8], and for a general closed njectve operator deal U n [9]. In every case a non-lnear verson of the factorzaton theorem of Daves, Fgel, Johnson and Pelczynsy (see [5, pgs. 250, 259]) through operators n U s obtaned for such multlnear mappngs. These results are then appled to get some factorzaton theorems for holomorphc mappngs. We are ready now to prove the followng characterzaton of the unform polymeasures. Theorem 2.4. Let K 1,..., K be compact Hausdorff spaces, let X be a Banach space and let T L (C(K 1 ),..., C(K ); X). Let γ : Σ 1 Σ X be the polymeasure assocated to t accordng to theorem 1.1. Then γ s unform f and only f T s regular. Besdes, n that case the measures φ defned before lemma 2.2 are the measures canoncally assocated to the operators T. Proof. Let us frst assume that γ s unform (n partcular ths means that γ s regular countably addtve and therefore X-valued, see [7]). Accordng to lemma 2.2 ths means that for each = 1,...,, φ rca(σ ; rcapm(σ 1,. [].. Σ ; X)). Snce rcapm(σ 1,. [].. Σ ; X) L 1 (C(K 1 ),. [].., C(K ); X) (cfr. theorem 1.1) we

4 4 FERNANDO BOMBAL AND IGNACIO VILLANUEVA get that φ rca(σ ; L 1 (C(K 1 ),. [].., C(K ); X)). Then we can consder the operator H φ L(C(K ); L 1 (C(K 1 ),. [].., C(K ); X)) assocated to φ by the Resz representaton theorem (vector valued case; see, f.. [5, Theorem VI.2.1]). Snce φ s countably addtve we now that H φ s wealy compact ([5, Theorem VI.2.5]). We consder now Hφ, the btranspose of H φ. Snce H φ s wealy compact we get that Hφ s L 1 (C(K 1 ),. [].., C(K ); X)-valued. It s easy to see that for every A Σ, and for every (f 1,. [].., f ) C(K 1 ). [].. C(K ), Hφ (A )(f 1,. [].., f ) = < φ (χ A ), (f 1,. [].., f ) >= (f 1,. [].., f )dγ A Therefore, = T (f 1,... f 1, χ A, f +1,... f ). Hφ (g )(f 1,. [].., f ) = T (f 1,... f 1, g, f +1,... f ), for every Σ -smple functon g and for every (f 1,. [].., f ) C(K 1 ). [].. C(K ). From contnuty, we get the same relaton for every g B(Σ ). In partcular, when we choose f C(K ) we get Hφ (f )(f 1,. [].., f ) = T (f 1,... f 1, f, f +1,... f ) = T (f 1,... f 1, f, f +1,... f ) = T (f )(f 1,. [].., f ). Obvously ths means that T = H φ and, therefore, that T s wealy compact. Let us now assume that T s regular. Then, for every = 1..., T L(C(K ); L 1 (C(K 1 ),. [].., C(K ); X) s wealy compact and so the measure µ assocated to t by the Resz representaton theorem s countably addtve and L 1 (C(K 1 ),. [].., C(K ); X)-valued ([5, Theorem VI.2.5]). We wll chec now that for every = 1..., µ = φ. Then, the proof wll be fnshed just by loong at lemma 2.2. Let T be the btranspose of T. For each A Σ let (f α) α I be a net n C(K ) such that f α ω χ A. T s nown to be ω -ω contnuous; beng T wealy compact we get that T s L 1 (C(K 1 ),. [].., C(K ); X)-valued. Both of these facts together mply that (T (f α)) α I converges wealy to T (χ A ). For fxed (f 1,. [].., f ) C(K 1 ). [].. C(K ) and x X, the lnear form θ : L 1 (C(K 1 ),. [].., C(K ); X) K defned as θ(s) := S(f 1,. [].., f ), x s clearly contnuous and therefore Besdes, θ(t θ(t (f α )) θ(t (χ A )) = T (χ A )(f 1,. [].., f ), x. (f α )) = T (f α )(f 1,. [].., f ), x = T (f 1,... f 1, f α, f +1,... f ), x. Snce T s separately ω -ω contnuous we get that ths last expresson converges to T (f 1,... f 1, χ A, f +1,..., f ), x. So we have obtaned that for every x X, T (f 1,... f 1, χ A, f +1,..., f ), x = T (χ A )(f 1,. [].., f ), x. Therefore for every A Σ and for every (f 1,. [].., f ) C(K 1 ) [] C(K ), T (f 1,... f 1, χ A, f +1,..., f ) = T (χ A )(f 1,. [].., f ) = µ (A )(f 1,. [].., f ).

5 REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 5 But clearly T (f 1,... f 1, χ A, f +1,..., f ) = (f 1,. [].., f )dγ A = φ (A )(f 1,. [].., f ). From here t follows that µ = φ and the proof s over. Snce every operator from C(K 1 ) to C(K 2 ) s wealy compact (cfr. [5, Theorem VI-2-15], f..), we get mmedately the followng result (see [6]): Corollary 2.5. Every regular countably addtve scalar bmeasure γ : Σ 1 Σ 2 K s unform. From the above theorem we can derve the followng propostons, useful to decde whether a polymeasure s or s not unform. Prevously we wll need a lemma. Lemma 2.6. Let T : C(K 1 ) C(K ) X be a regular - lnear operator. Let (f n) n N C(K ) be a wealy null sequence and let ((g1 n ) n N,. [].., (g n) n N) B(Σ 1 ). [].. B(Σ ) be bounded sequences. Then, wth the notaton of theorem 1.1, T (g1 n,... g 1 n, f n, gn +1,..., gn ) converges n norm to zero. Proof. If T s regular, then the above defned operator T s wealy compact and therefore completely contnuous, by the Dunford-Petts property of C(K ). Ths means that T (f n ) 0. We observe now that, due to the unqueness of the extenson (1.1), for every (g 1,. [].., g ) B(Σ 1 ). [].. B(Σ ) and for every f C(K ), we have T (f )(g 1,. [].., g ) = T (g 1,... g 1, f, g +1,..., g ). By the equalty of the norms of the operator and ts extenson, we can wrte T (f n ) 0. Ths can also be wrtten as whch means that and fnshes the proof. sup T (f n)(g 1,. [].., g ) 0, g j B B(Σj ) sup T (g 1,... g 1, f n, g +1,..., g ) 0 g j B B(Σj ) Proposton 2.7. A regular countably addtve polymeasure γ : Σ 1 Σ X s unform n the ı th varable f and only f the measures {γ g1 : (g,...,g [] 1,. [].., g ) B(Σ 1 ) [] B(Σ ), g j 1} are unformly countably addtve. Proof. One of the mplcatons s clear. For the other, let us suppose that γ s unform n the ı th varable. Were the measures {γ g1 ; (g,...,g [] 1,. [].., g ) B(Σ 1 ) [] B(Σ )} not unformly countably addtve, then there would exst ɛ > 0, a sequence (A n ) n N Σ of dsjont open sets and sequences ((g1 n ) n N,. [].., (g n) n N) B(Σ 1 ). [].. B(Σ ) wth gj n [] 1 for each n N and for each j = 1..., such that γ g1, (A n...,g [] ) > ɛ. Then for each n N there would exst f n C(K ) wth suppf n A n and f n 1 such that f ndγ > ɛ, and ths n contradcton wth lemma 2.6, snce the sequence f n converges wealy to 0. g 1,...,g []

6 6 FERNANDO BOMBAL AND IGNACIO VILLANUEVA Proposton 2.8. A regular countably addtve polymeasure γ : Σ 1 Σ X s unform n the ı th varable f and only f the measures {γ f1 ; (f,...,f [] 1,. [].., f ) C(K 1 ) [] C(K ), f j 1} are unformly countably addtve. Proof. In one drecton the result follows from the prevous proposton. For the other, we wll suppose wthout loss of generalty that =. Let us suppose that the measures {γ f1,...,f 1 ; (f 1,..., f 1 ) C(K 1 ) C(K 1 ), f j 1} are unformly countably addtve. If γ s not unform n the th varable then there exst a sequence A n Σ of dsjont open sets and sequences (A n j ) n N Σ j for j = such that γ(a n 1,..., A n, ) > ɛ. Snce γ s regular, γ(, An 2,... A n ) s regular for each n N and therefore there exsts a functon f1 n C(K 1 ) wth f1 n 1 such that f1 n dγ A n 2,...,A n > ɛ. Now γ f1 n,,χ A n,...,χ s also regular and therefore there exsts a functon f2 n C(K 2 ) wth f2 n 1 such that 3 A n f2 n dγ f n 1,χ A n,...,χ 3 A n > ɛ. Contnung n the same way we obtan 1 sequences of norm one functons fj n C(K j), j = such that γ f n 1,...,f 1 n (An ) > ɛ whch contradcts the hypothess. 3. Polymeasures on compact dspersed spaces Recall that a compact Hausdorff space s sad to be dspersed f t does not contan any non empty perfect set. In [12] a deep nsght s gven nto the structure of dspersed spaces, provng among other results that K s dspersed f and only f C(K) contans no copy of l 1, f and only f C(K) contans no copy of L 1. Also, n ths case C(K) can be dentfed wth l 1 (Γ) for some Γ. Some (f not all) of the followng results are probably nown, but we have not been able to fnd an explct reference. Theorem 3.1. For a compact Hausdorff space K, the followng statements are equvalent: a) K s dspersed. b o ) For every 1, the space L (C(K)) s Schur. b 1 ) For some 2, the space L (C(K)) s Schur. b 2 ) For some 2, the space P( C(K)) s Schur. b 3 ) For every 2, the space P( C(K)) s Schur. c o ) For every 1, the space L (C(K)) s wealy sequentally complete. c 1 ), c 2 ), c 3 ): Same statements as b 1 ), b 2 ), b 3 ), replacng Schur by wealy sequentally complete. d o ) For every 1, L (C(K)) contans no copy of l. d 1 ), d 2 ), d 3 ): Same statements as b 1 ), b 2 ), b 3 ), replacng Schur by the non contanment of l. e) For every 1, L (C(K)) contans no copy of c o. e 1 ), e 2 ), e 3 ): Same statements as d 1 ), d 2 ), d 3 ), replacng l by c o. Proof. Snce L (C(K)) s a dual space for every 1, every (d) statement s equvalent to the correspondng (e) statement. Also clearly b ) c ) d ), for every, b o ) b 1 ) b 2 ) and b o ) b 3 ) b 2 ). Therefore, t rests to prove a) b o ) and e 2 ) a).

7 REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 7 a) b o ): We shall prove t by nducton on. For = 1, t s clear snce C(K) l 1 (Γ). Suppose now that ( ) L (C(K)) = C(K) := X π (cfr. [5, Corollary VIII.2.2]) s Schur. Then L +1 (C(K)) = L(C(K); X ) = ( C(K) ˆ π X ). Snce C(K) contans no copy of l 1 and has the Dunford-Petts property, by the nducton hypothess t follows that all members of the last space are compact operators. Hence, snce C(K) has the approxmaton property, L +1 (C(K)) = C(K) ˆ ɛ X ([5, Theorem VIII.3.6]), whch s a Schur space, snce ths property s stable by tang njectve tensor products (cfr. f.. [13]). e 2 ) a): If K s not dspersed, C(K) L 1 l 2. Consequently l 2 ˆ ɛ l 2 C(K) ˆ ɛ C(K) ( C(K) ˆ π C(K) ) (topologcal nclusons), and t s well nown that f (e n ) s the canoncal bass of l 2, then (e n e n ) s equvalent to the canoncal bass of c o (cfr., f.., [10]). Ths means that P( 2 C(K)) contans a copy of c o. Snce P( 2 C(K)) s a (complemented) subspace of P( C(K)), for every 2, t follows that the latter space contans a copy of c o, too. As we menton n corollary 2.5, every scalar regular bmeasure on a compact Hausdorff space s always unform. Ths s not true for arbtrary polymeasures, as the followng example from [2] shows: The 3-lnear map T : C([0, 1]) C([0, 1]) C([0, 1]) C defned by T (f, g, h) := f( ) gr dx hr dx, =1 where r s the standard th Rademacher functon, s not regular. See [2] for detals. In the next theorem we show that the unformty of all the -polymeasures for some (every) 3, characterzes the compact dspersed spaces. We shall denote by K(X; Y ) and W(X; Y ) the compact and wealy compact operators between X and Y, respectvely. Theorem 3.2. For a compact Hausdorff space K the followng statements are equvalent a) K s dspersed. f) For every (some) 2, L ( C(K); L (C(K)) ) = K ( C(K); L (C(K)) ). g) For every (some) 2, L ( C(K); L (C(K)) ) = W ( C(K); L (C(K)) ). h) For every (some) 3, any scalar regular -polymeasure on the product of the Borel σ-algebra of K, s unform. Proof. a) f) was ncluded n the proof of a) b o ) n theorem 3.1, and clearly f) g). The equvalence of (g) and (h) follows from theorem 2.4. Fnally, let us prove that (g) mples (a): Let 3. If K s not dspersed, C(K) s nfnte dmensonal and thus contans a copy of c o ([5, Corollary VI.2.16]). On the other hand, by theorem 3.1, L 1 (C(K)) contans a copy of l. By the njectvty of ths 0 0

8 8 FERNANDO BOMBAL AND IGNACIO VILLANUEVA space, the ncluson map from c o nto l can be extended to the whole space C(K), provdng n ths way a non wealy compact operator n L ( C(K); L 1 (C(K)) ). The equvalence of (a), (f) and (g) has been also obtaned n [1], although wth a dfferent and, n our opnon, more nvolved proof. References [1] Alamnos, J., Cho, Y. S., Km, S. G. and Pay, R., Norm attanng blnear forms on spaces of contnuous functons. Preprnt. [2] Aron, R., Cho, S. Y. and Llavona, J. L. G., Estmates by polynomals, Bull. Austral. Math. Soc. 52 (1995), [3] Aron, R. and Galndo, P., Wealy compact multlnear mappngs, Proc. of the Ednburgh Math. Soc., 40 (1997) [4] Bombal, F. and Vllanueva, I., Multlnear operators n spaces of contnuous functons, to appear. [5] Destel, J. and Uhl, J. J., Vector Measures, Mathematcal Surveys, No. 15. Amercan Math. Soc., Provdence, R.I., [6] Dobraov, I., On ntegraton n Banach spaces, VIII (polymeasures), Czech. Math. J. 37 (112) (1987), [7] Dobraov, I., Representaton of multlnear operators on C 0 (T ), Czech. Math. J. 39 (114) (1989), [8] Gonzlez, M. and Gutrrez, J.,Factorzaton of wealy contnuous holomorphc mappngs, Studa Math., 118 (2) (1996), [9] González, M. and Gutérrez, J., Injectve factorzaton of holomorphc mappngs, To appear. [10] Holub, J.R., Tensor product bases and tensor dagonals, Trans. A.M.S. 151 (1970), [11] Pelczyns, A., A theorem of Dunford- Petts type for polynomal operators, Bull. Acad. Polon. Sc. Ser. Sc. Math. Astr. Phys., 11 (1963), [12] Pelczynsy, A. and Semaden, Z., Spaces of contnuous functons (III). Spaces C(Ω) for Ω wthout perfect subsets, Studa Math. XVIII (1959), [13] Ruess, W. and Werner, D., Structural propertes of operator spaces, Acta Unv. Carol. 28 (1987), [14] Vllanueva, I., Polmeddas y representacón de operadores multlneales de C(Ω 1, X 1 ) C(Ω d, X d ). Tesna de Lcencatura, Dpto. de Análss Matemátco, Fac. de Matemátcas, Unversdad Complutense de Madrd, 1997.