2000 Mathematics Subject Classification: 46B50, 46B70, 46G12.A new measure of noncompactness on Banach spaces is defined from the Hausdorff measure of noncompactness, giving a quantitative version of a classical result by R. S. Phillips. From the main result, classical results are obtained now as corollaries and we have an application to interpolation theory of Banach spaces

da Silva, Eduardo Brandani

Fernanadez, Dicesar L.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 32 (2006), 179–184HAUSDORFF MEASURES OF NONCOMPACTNESS ANDINTERPOLATION SPACESEduardo Brandani da Silva, Dicesar L. FernanadezCommunicated by S. L. TroyanskiAbstract. A new measure of noncompactness on Banach spaces is definedfrom the Hausdorff measure of noncompactness, giving a quantitative versionof a classical result by R. S. Phillips. From the main result, classical resultsare obtained now as corollaries and we have an application to interpolationtheory of Banach spaces.Introduction. The notion of measure of noncompactness was introdu-ced by K. Kuratowski and, with a convenient but equivalent modification, by F.Hausdorff. Subsequently it was used in numerous branches of functional analysisand theory of differential and integral equations. In this note we introduce a newmeasure of noncompactness to obtain a quantitative version of a classical resultby R. S. Phillips [5, Thm. 3.7] (see also Dunford-Schwartz [4, Lemma IV.5.4, p.259] and Brooks-Dinculeanu [3, Thm. 1]). We shall also give an application tointerpolation spaces.2000 Mathematics Subject Classification: 46B50, 46B70, 46G12.Key words: Banach spaces, measures of noncompactness, interpolation.180 Eduardo Brandani da Silva, Dicesar L. Fernanadez1. Hausdorff measures of noncompactnes. Given a Banach spaceX, the closed unitary ball in X is denoted by UX . The Haudorff measure ofnoncompactness of a bounded subset B ⊂ X is defined byχX(B) = inf{ε > 0 : there exists a finite set F in X such that B ⊂ F + εUX }.For properties of χ see [1].2. A Phillips-like estimate. We shall state a quantitative version,but sligthly more general, of Brooks-Dinculeanu’s Theorem 1 [3].If (Xn), n ∈ N is a sequence of Banach spaces, for 1 ≤ p < ∞, we denotebyXp = p∞⊕n=1Xn,the Banach space of all sequences (xn) in∞∏n=1Xn such that‖(xn)‖Xp =[∞∑n=1‖xn‖pXn]1/p< ∞.Given a sequence (xn) in Xp, let us set Pk(xn) = (x1, . . . , xk, 0, 0, . . .) andpik((xn)) = xk, the projection on the kth-component.Theorem 2.1. For a bounded subset B ⊂ p⊕∞n=1Xn we setν(B) = lim supk→∞[supx∈B‖Pk(xn)− (xn)‖Xp + χ(Pk(B))].Then, if χ is the Hausdorff measure of noncompactness in Xp, we haveχ(B) ≤ ν(B) ≤ 2 χ(B),for all bounded subset B in Xp.P r o o f. For each bounded subset B ⊂ Xp and n ∈ N, we haveB ⊂ (Id− Pn)B + PnB.Since the Hausdorff measure of noncompactness is subadditive, taking in accountthe inequalityχ((Pn − Id)B) ≤ supx∈B||Pnx− x||,Hausdorff measures of noncompactness and interpolation spaces 181we getχ(B) ≤ χ((Pn − Id)B) + χ(PnB)≤ supx∈B||Pnx− x||+ χ(PnB),for all n ∈ N. Therefore, χ(B) ≤ ν(B).Conversely, since operators Pn are uniformly bounded, let us define M :=lim supn→∞ ‖Pn‖. Then, since ‖Pn‖ = 1 (‖Pnx‖ ≤ ‖x‖ for all x and ‖Pnx‖ = ‖x‖for x = (x1, . . . , xn, 0, 0, . . .)), it follows M = 1.Given a bounded fixed subset B in X, let r = χ(B) and, for ε > 0arbitrary, let rε := r + ε. Thus, there is a finite set B0 in X such thatB ⊂ B0 + rεUXp .And, since B0 is finite, there exist N ∈ N such that‖(Pn − Id)x0‖ < ε,for all n ≥ N and x0 ∈ B0. Now, let x an arbitrary element in B and x0 ∈ B0chosen such that ‖x− x0‖ < rε. Since‖(Pn − Id)x‖ − ‖(Pn − Id)x0‖ ≤ ‖(Pn − Id)(x− x0)‖ ≤ 1 . rε,it holds‖(Pn − Id)x‖ ≤ ‖(Pn − Id)x0‖+ rε,and, for all x ∈ B and n ≥ N , we have‖Pnx− x‖ ≤ ε + rε = r + 2ε.Therefore, taking ε → 0 one haslim supn→∞supx∈B||Pnx− x|| ≤ χ(B).Finally, since χ(PλB) ≤ ‖Pn‖ χ(B) ≤ M χ(B) ≤ χ(B) we getν(B) ≤ χ(B) + χ(B) = 2χ(B),and the proof is complete. From the result of the Theorem 2.1 we can prove the measure ν has allthe properties of χ, therefore ν is a measure of noncompactness too. And albeit νis a measure equivalent to χ, from ν we get the new results which follows below.182 Eduardo Brandani da Silva, Dicesar L. FernanadezThe next result is necessary to get our main application.Lemma 2.2. For a bounded subset B ⊂ Xp = p⊕∞n=1Xn we haveχXn(pin(B)) ≤ χ(B).P r o o f. We start verifying that pin(UXp) = UXn . Let x = (xj)∞j=1 ∈ UXp ,then‖x‖Xp = ∞∑j=1‖xj‖pXj1/p ≤ 1.Thus, we have ‖xj‖Xj ≤ ‖x‖Xp ≤ 1 for all j. Since xn = pin(x) we obtainpin(x) ∈ UXn and finally pin(UXp) ⊂ UXn . Now, given z ∈ UXn , we define asequence x = (xj)∞j=1 by xj = 0, if j 6= n, and xj = z, if j = n. Then x ∈ Xpand ‖x‖Xp = ‖z‖Xn ≤ 1, which implies x ∈ UXp and pin(x) = z. Therefore, givenz ∈ Xn, there exists x ∈ Xp with pin(x) = z, what means UXn ⊂ pin(UXp) andthe assertion follows.Now, given ε > χ(B), there exist balls B1, . . . , BM ∈ Xp which Bi =B(xi, ε), such thatB ⊂M⋃i=1B(xi, ε).Thus,pin(B) ⊂ pin(M⋃i=1B(xi, ε))⊂M⋃i=1pin(B(xi, ε)).Now, sincepin(B(xi, ε)) = pin(xi) + εpin(UXp) = pin(xi) + εUXn ,for each i, we see that there exist elements y1, . . . , yM such thatpin(B) ⊂M⋃i=1{yi + εUXn)}.Therefore, χXn(pin(B)) ≤ ε and the result follows. Corollary 2.3. A set K ⊂ Xp = p⊕∞n=1Xn is relatively compact, if andonly if:A)∑m≥k||xm||pXn−→ 0, k →∞, uniformly for x ∈ K.Hausdorff measures of noncompactness and interpolation spaces 183B) the set K(m) = {xm = pim(x) ; x ∈ K} is relatively compact in thenorm of Xm, for each m ∈ N.P r o o f. If K ⊂ Xp is relatively compact, we have χ(K) = 0 and, byTheorem 2.1, we obtainν(K) = lim supk→∞[supx∈B‖Pk(xn)− (xn)‖Xp + χ(Pk(K))]= 0.From Lemma 2.2, we have for each nχXn(pin(K)) = χXn(pin(Pn(K))) ≤ χ(Pn(K)) ≤ ‖Pn‖L(Xp,Xp) χ(K),thus, A) and B) follow. In particular, if X is a fixed Banach space and Xn = X, for each n ∈ N,we haveXp = p∞⊕n=1Xn = `pX .Thus, we obtain from Corollary 2.3 a result stated by Brooks-Dinculeanu [1,Thm. 1].Corollary 2.4. A set K ⊂ `pX , 1 ≤ p < ∞, is relatively compact, if andonly if:A)∑m≥k‖xm‖p −→ 0, k →∞, uniformly for x ∈ K.B) for each m ∈ N, the set K(m) = {xm;x ∈ K} is relatively compactin the norm of X.3. An application to interpolation spaces. Given a Banach spaceE and a number α > 0, we set αE for the space E equipped with the norm‖ . ‖αE = α ‖ . ‖E .Let (E0, E1) be a Banach couple and 0 < θ < 1 (see [2] for the definitionson interpolation theory of Banach spaces). For each n ∈ Z we setXθn := 2−θnE0 + 2−(θ−1)nE1.For 1 ≤ p < ∞, the K-interpolation space (E0, E1)θ,p,K can be identified with thesubspace of all constant sequences in p⊕̂n∈ZXθn. Then, for each n ∈ N, settingIn for the segment in Z from −n to n and Icn = Z \ In, we see that the functionalνθ(B) := lim supn→∞[supx∈B[∑k∈Icn[2−kθK(2k, x)]p]1/p + χ(PIn(B))]184 Eduardo Brandani da Silva, Dicesar L. Fernanadezcan be estimate in (E0, E1)θ,p,K .As a consequence of the main theorem, we have the following compactnesscriterion for bounded sets in interpolation spaces, which goes back to J. Peetre.Theorem 3.2 (J.Peetre). Let (E0, E1)θ,p,K be an interpolation space with0 < θ < 1 and 1 ≤ p < ∞. Then, a bounded subset B in (E0, E1)θ,p,K is relativelycompact if and only ifA) lim supn→∞∑k∈IcN[2−kθK(2k, x)]p = 0, uniformly in x ∈ B,andB) the subset B is relatively compact in E0 + E1.Indeed, νθ(B) can be estimate by the Hausdorff measure of noncompact-ness χ(B). Further, if B is precompact in E0 + E1 is also precompact in Xθn =2−θnE0 + 2−(θ−1)nE1.REFERE NCES[1] J. Banas´, K. Goebel. Measures of Noncompactness in Banach Spaces.Marcel Dekker Inc. New York, 1980.[2] J. Bergh, J. Lo¨fstro¨n. Interpolation Spaces: An Introduction. Springer-Verlag, 1976[3] J. K. Brooks, N. Dinculeanu. Conditional expectation and weak andstrong compactness in spaces of Bochner integrable functions. J. Mult.Analysis 9 (1979), 420–427.[4] N. Dunford, J. Schwartz. Linear Operators. Part I. Interscience Pub.Inc. New York, 1967.[5] R. S. Phillips. On linear transformations. Trans. Amer. Math. Soc. 48(1940), 516-541.Eduardo Brandani da SilvaDepto. de Mat. UEM. Maringa´ PR Brasile-mail: ebsilva@uem.brDicesar L. FernanadezIMECC-UNICAMP Campinas, Brasile-mail: dicesar@ime.unicamp.brReceived November 11, 2005Revised January 26, 2006

Hausdorff Measures of Noncompactness and Interpolation Spaces

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Hausdorff Measures of Noncompactness and Interpolation Spaces

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