We revised the known results on interpolation of the measure of noncompactness and we announce a new approach to establishing the interpolation formula for the real method

Cobos, Fernando

Fernández-Cabrera, Luz M.

Martínez, Antón

We revised the known results on interpolation of the measure of noncompactness and we announce a new approach to establishing the interpolation formula for the real method. Key words: real interpolation, measure of noncompactness

Fernando Cobos

Luz M. Fernández-cabrera

Antón Martínez

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On interpolation of the measure of noncompactness

We revised the known results on interpolation of the measure of noncompactness and we announce a new approach to establishing the interpolation formula for the real method.Ministerio de Educación, Formación Profesional y Deportes (España)Comunidad de MadridUniversidad Complutense de MadridDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Cobos Díaz, Fernando

Fernández-Cabrera Marín, Luz María

Docta Complutense

Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 3, 159–163On interpolation of the measure ofnoncompactnessFernando Cobosa, Luz M. Fernández-Cabrerab, and Antón Martínezca Departamento de Análisis Matemático, Facultad de Matemáticas, UniversidadComplutense de Madrid, 28040 Madrid, Spain; cobos@mat.ucm.esb Sección Departamental de Matemática Aplicada, Escuela de Estadística, UniversidadComplutense de Madrid, 28040 Madrid, Spain; luz_fernandez-c@mat.ucm.esc Departamento de Matemática Aplicada I, E.T.S. Ingenieros Industriales, Universidad deVigo, 36200 Vigo, Spain; antonmar@uvigo.esReceived 10 April 2006Abstract. We revised the known results on interpolation of the measure of noncompactnessand we announce a new approach to establishing the interpolation formula for the real method.Key words: real interpolation, measure of noncompactness.1. INTRODUCTION AND PRELIMINARIESLet Ā = (A0, A1) be a Banach couple, i.e., two Banach spaces Aj , j = 0, 1,which are continuously embedded in a common Hausdorff topological vectorspace, and letK(t, a) = inf{‖a0‖A0 + t‖a1‖A1 : a = a0 + a1, aj ∈ Aj}, a ∈ A0 + A1,andJ(t, a) = max{‖a‖A0 , t‖a‖A1}, a ∈ A0 ∩A1,be Peetre’s K- and J-functionals. As it is well known, for 0 < θ < 1 and1 ≤ q ≤ ∞, the real interpolation space (A0, A1)θ,q, realized in a discrete wayas a K-space, is formed by all those a ∈ A0 + A1 which have a finite norm‖a‖θ,q;K =( ∞∑m=−∞(2−θmK(2m, a))q)1/q159(the sum should be replaced by the supremum if q = ∞). We denote by Āθ,q;K thespace (A0, A1)θ,q normed by ‖ · ‖θ,q;K . This space can be defined equivalently byusing the J-functional. It turns out that a belongs to (A0, A1)θ,q if it can be writtenas a sum a =∑∞m=−∞ um (convergence in A0 +A1), where {um} ⊆ A0∩A1 and( ∞∑m=−∞(2−θmJ(2m, um))q)1/q< ∞. (1.1)Moreover, the infimum of all values (1.1) over all possible representations of a asabove defines a norm ‖a‖θ,q;J which is equivalent to ‖ · ‖θ,q;K . We write Āθ,q;J forthe space (A0, A1)θ,q normed by ‖ · ‖θ,q;J . By Āθ,q = (A0, A1)θ,q we mean any ofthe spaces Āθ,q;K or Āθ,q;J . Full details on the real method can be found in [1,2].Let B̄ = (B0, B1) be another Banach couple, and let T ∈ L(Ā, B̄), that is, T isa linear operator from A0 +A1 into B0 +B1 whose restriction to each Aj defines abounded operator from Aj into Bj for j = 0, 1. It is well known that the restrictionT : (A0, A1)θ,q −→ (B0, B1)θ,q is bounded and the following inequality holds:‖T‖Āθ;q ,B̄θ;q ≤ C ‖T‖1−θA0,B0‖T‖θA1,B1 .Similar formulae have been established for other quantities (see [3−5]), but nowsome of them require much effort to be proved. This is the case of the measure ofnoncompactness, a concept that means more than only continuity but not so muchas compactness.Given a bounded linear operator T ∈ L(A,B), between the Banach spacesA and B, we recall that the measure of noncompactness β(T ) = β(TA,B) ofT is defined as the infimum of all σ > 0 such that there exists a finite subset{b1, . . . , bs} ⊆ B so thatT (UA) ⊆s⋃j=1{bj + σUB}.Here, UA stands for the closed unit ball of A. The symbol UB has a similarmeaning. It is clear that β(TA,B) ≤ ‖T‖A,B , and T is compact if and only ifβ(T ) = 0 (see [6] for more details on this notion).In 1999, Fernández-Martínez and two of the present authors established thefollowing interpolation formula for the measure of noncompactness:β(TĀθ;q ,B̄θ;q) ≤ C β(TA0,B0)1−θβ(TA1,B1)θ. (1.2)The proof is based on the vector-valued sequence spaces that come up whendefining the real interpolation space. Using them and certain families of projec-tions, one can decompose and approximate the operator T by other operatorswhose measure of noncompactness can be directly estimated. This technique has160its origin in the papers [7−10]. Previous results on interpolation of the measure ofnoncompactness are due to Edmunds and Teixeira [11]. The results of [11] referto the special case when A0 = A1, or B0 = B1, or the couple (B0, B1) satisfiesa certain approximation condition. Under these extra assumptions, the proofs areeasier and results also apply to the complex method. However, it is an open problemif a similar formula to (1.2) holds for the complex method.The aim of this note is to announce a new approach to (1.2), which simplifiesthe way of handling the “middle parts” of the operator. The main lines of the proofare described in the next section. Full details can be found in the paper [12].2. SKETCH OF THE MAIN LINES OF THE PROOFIn order to establish (1.2), we realize Āθ,q as a J-space and B̄θ,q as aK-space. This will allow us to work with the vector-valued sequence spaceswhich are implicit in those descriptions. Namely, for m ∈ Z, we denote byGm the Banach space A0⋂A1 endowed with the norm J(2m, ·), and by Fmwe mean B0 + B1 normed by K(2m, ·). Then we consider the vector-valuedsequence spaces `q(2−θmGm) and `q(2−θmFm). The relationship between thefirst space and Āθ,q;J is given by the metric surjection π{um} =∑m∈Z um. Theconnection between B̄θ,q;K and `q(2−θmFm) is given by the isometric embeddingιb = (. . . , b, b, b, . . . ). The relevant diagram to keep in mind is`1(Gm)π−→ A0 T−→ B0 ι−→ `∞(Fm)`1(2−mGm)π−→ A1 T−→ B1 ι−→ `∞(2−mFm)`q(2−θmGm)π−→ Āθ,q;J T−→ B̄θ,q;K ι−→ `q(2−θmFm).Besides π and ι the following sequences of operators will be useful forestablishing (1.2):Pn{um} = {. . . 0, u−n, . . . , un, 0, . . .}, Q+n {um} = {. . . 0, 0, un+1, un+2, . . .}, andQ−n = I − Pn −Q+n . They satisfy‖Q+n ‖`1(Gm),`1(2−mGm) = 2−(n+1) = ‖Q−n ‖`1(2−mGm),`1(Gm)and‖Pn‖`1(Gm),`1(2−mGm) = 2n = ‖Pn‖`1(2−mGm),`1(Gm).Similar equalities hold when these operators are considered acting on the couple of`∞-spaces modelled on {Fm}.Since π is a metric surjection, we haveβ(TĀθ,q;J ,B̄θ,q;K ) = β(Tπ`q(2−θmGm),B̄θ,q;K ) ≤ β(TπP2n) + β(Tπ(Q+2n + Q−2n)).161To deal with the first term, we use a functional which is equivalent to β. GivenS ∈ L(A,B), let γ(S) be the infimum of the set of all σ > 0 for which there is aBanach space Z and a compact linear operator R ∈ L(A, Z) such that‖Sx‖B ≤ σ‖x‖A + ‖Rx‖Z for all x ∈ A.It turns out that 12β(S) ≤ γ(S) ≤ 2β(S). This equivalence allows us to establishthatβ(TπP2n) ≤ C β(TA0,B0)1−θβ(TA1,B1)θ for all n ∈ N.For the other terms, we haveβ(Tπ(Q+2n + Q−2n)) ≤2β((Pn + Q+n + Q−n )ιTπ(Q+2n + Q−2n))≤2[β(PnιTπQ+2n) + β(PnιTπQ−2n)+ β(Q+n ιTπQ−2n) + β(Q−n ιTπQ+2n)+ β(Q+n ιTπQ+2n) + β(Q−n ιTπQ−2n)],where the last six operators act from `q(2−θmGm) into `q(2−θmFm). We estimateeach of these terms by the norm of the corresponding operator. Using properties ofPn, Q+n , Q−n , it follows that given any ε > 0, there is N ∈ N such thatβ(Tπ(Q+2n + Q−2n)) ≤ (1 + ε)Cβ(TA0,B0)1−θβ(TA1,B1)θ for any n ≥ N.Full details can be found in [12]. It is also shown there that this approachworks to estimate the measure of noncompactness of operators interpolated by thegeneral real method (A0, A1)Γ. The spaces (A0, A1)Γ are defined by replacing theweighted `q-norm of (A0, A1)θ,q by a more general lattice norm (see [13]).ACKNOWLEDGEMENTSThis research was supported in part by the Ministry of Education and Scienceof Spain (project MTM2004-01888) and CAM-UCM (Research group 910348).F. Cobos was partly supported by RTM PHD FP6-511953.REFERENCES1. Bergh, J. and Löfström, J. Interpolation Spaces. An Introduction. Springer, Berlin, 1976.2. Triebel, H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978.3. Cobos, F., Manzano, A. and Martínez, A. Interpolation theory and measures related tooperator ideals. Quart. J. Math., 1999, 50, 401–416.4. Cobos, F. and Martínez, A. Remarks on interpolation properties of the measure of weaknon-compactness and ideal variations. Math. Nachr., 1999, 208, 93–100.1625. Cobos, F., Fernández-Martínez, P. and Martínez, A. Interpolation of the measure of non-compactness by the real method. Studia Math., 1999, 135, 25–38.6. Schechter, M. Principles of Functional Analysis. American Mathematical Society,Providence, 2002.7. Cobos, F., Edmunds, D. E. and Potter, A. J. B. Real interpolation and compact linearoperators. J. Funct. Anal., 1990, 88, 397–401.8. Cobos, F. and Fernandez, D. L. On interpolation of compact operators. Ark. Mat., 1989,27, 211–217.9. Cobos, F. and Peetre, J. Interpolation of compactness using Aronszajn–Gagliardofunctors. Israel J. Math., 1989, 68, 220–240.10. Cobos, F., Kühn, T. and Schonbek, T. One-sided compactness results for Aronszajn–Gagliardo functors. J. Funct. Anal., 1992, 106, 274–313.11. Teixeira, M. F. and Edmunds, D. E. Interpolation theory and measure of non-compactness.Math. Nachr., 1981, 104, 129–135.12. Cobos, F., Fernández-Cabrera, L. M. and Martínez, A. Abstract K and J spaces andmeasure of non-compactness. Math. Nachr. (to appear).13. Nilsson, P. Reiteration theorems for real interpolation and approximation spaces. Ann.Mat. Pura Appl., 1982, 132, 291–330.Mittekompaktsuse mõõdu interpoleerimineFernando Cobos, Luz M. Fernández-Cabrera ja Antón MartínezOn revideeritud tuntud tulemusi mittekompaktsuse mõõdu interpoleerimisekohta. On esitatud uus lähenemine interpolatsiooni valemi tõestamiseks reaalsemeetodi puhul.163

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On interpolation of the measure of noncompactness

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