We show that the Banach–Saks property with respect to a regular positive matrix method of summability is inherited by the real interpolation spaces from a space forming the interpolation family and possessing this property. The proof refers to the Galvin–Prikry theorem on Ramsey sets. The results apply to several matrix methods of summability, such as Cesaro, Nørlund or Holder methods

Kryczka, Andrzej

Kurlej, Konrad

Annales Universitatis Mariae Curie-Sklodowska sectio A – Mathematica

English

Biblioteka Nauki - repozytorium artykuÅÃ³w

Regular matrix methods of summability and real interpolation

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

doi: 10.17951/a.2021.75.1.53-59ANNALESU N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K AL U B L I N – P O L O N I AVOL. LXXV, NO. 1, 2021 SECTIO A 53–59ANDRZEJ KRYCZKA and KONRAD KURLEJRegular matrix methods of summabilityand real interpolationAbstract. We show that the Banach–Saks property with respect to a regu-lar positive matrix method of summability is inherited by the real interpola-tion spaces from a space forming the interpolation family and possessing thisproperty. The proof refers to the Galvin–Prikry theorem on Ramsey sets.The results apply to several matrix methods of summability, such as Cesàro,Nørlund or Hölder methods.1. Introduction. One of the fundamental questions in interpolation the-ory is whether a property of Banach spaces forming the interpolation coupleor family is inherited by interpolation spaces. A similar question can beposed for interpolated operators. The main results in the literature con-cern reflexivity, weak and strong compactness, or more general, propertiesrelated to closed operator ideals (see [10]).The Banach–Saks property lies within reflexive spaces and, respectively,weakly compact operators. The interpolation theorem for the classicalBanach–Saks property, that is, defined by arithmetic means, was obtainedby Beauzamy in [1], where the main tool used in the proof is Rosenthal’sresult, closely related to spreading models.In our paper we propose a different approach to this problem. Instead ofspreading models, we refer more directly to the Galvin–Prikry theorem on2010 Mathematics Subject Classification. 40C05, 40G05, 46B70.Key words and phrases. Real interpolation spaces, regular methods of summability,Banach–Saks property.54 A. Kryczka and K. KurlejRamsey sets [4]. We apply a result concerning behavior of regular summa-tion for direct sums of Banach spaces recently proved in [6]. We obtain ageneral result for all positive matrix methods of summability which includethe lower triangular matrix of arithmetic means. We also extend results tothe interpolation of finite families of Banach spaces developed by Sparr [11]and Yoshikawa [12].2. Real interpolation. In interpolation of Banach spaces, passing fromtwo to several spaces brings some difficulties. If n > 1, the interpolationspaces defined by J- and K-methods for Banach (n+1)-tuples are no longerequivalent (see [11, Remark 5.1]). Nevertheless, the inheritance of someproperties is similar for couples and families of Banach spaces. We showthat this is the case for regular summation and spaces defined by the K-method. We briefly recall basic facts on interpolation of finite families ofBanach spaces (for details see [11], [12]).Let n ≥ 1 and A0, . . . , An be Banach spaces, linearly and continuouslyembedded in a Hausdorff topological vector space V . Then ~A = (A0, . . . , An)is called a Banach (n+ 1)-tuple. The spaces∆( ~A) = A0 ∩ . . . ∩An, Σ( ~A) = A0 + . . .+Anare Banach spaces with norms, respectively,‖a‖∆( ~A) = max0≤k≤n ‖a‖Ak , ‖a‖Σ( ~A) = infn∑k=0‖ak‖Ak ,where the infimum is taken over all decompositionsa = a0 + . . .+ anwith ak ∈ Ak for k = 0, . . . , n.LetRn+ = {(t1, . . . , tn) ∈ Rn : tk > 0, k = 1, . . . , n}andHn+1+ ={(θ0, . . . , θn) ∈ Rn+1 :n∑k=0θk = 1, θk > 0, k = 0, . . . , n}.For θ = (θ0, . . . , θn) ∈ Hn+1+ let ξk = (ξk1 , . . . , ξkn) be given by the formulaξki ={θi − 1 if k > 0 and i = kθi otherwise,where i = 1, . . . , n and k = 0, . . . , n.For a ∈ Σ( ~A), t = (t1, . . . , tn) ∈ Rn+ and 1 ≤ q <∞ we putKq(t, a) = inf(‖a0‖qA0 +n∑k=1tqk ‖ak‖qAk)1/q,Regular matrix methods of summability... 55where the infimum is taken as for ‖a‖Σ( ~A).Let us now fix 1 ≤ p <∞. The Sparr real interpolation space Aθp;K withrespect to ~A consists of all a ∈ Σ( ~A) for which the norm(1) ‖a‖θp;K =(∫Rn+(t−θ11 . . . t−θnn K1(t, a))p dt1t1. . .dtntn)1/pis finite. The space Aθp;K is a Banach space such that the inclusions∆( ~A) ⊂ Aθp;K ⊂ Σ( ~A)are continuous. This construction of interpolation spaces is called the K-method.The Sparr space can also be defined by a discrete norm. Given v =(v1, . . . , vn) ∈ Zn, we write 2v = (2v1 , . . . , 2vn). For vectors ξ ∈ Rn andv ∈ Zn let ξ · v denote its standard scalar product. Then the norm‖a‖θp;Kq =(∑v∈Zn(2−ξ0·vKq(2v, a))p)1/pis equivalent to that of (1) if we put q = 1 (see [11, Remark 4.5]). Moreover,the norms ‖·‖θp;Kq and ‖·‖θp;Kr are equivalent for all 1 ≤ q, r <∞.Our results in this paper depend on the behavior of regular summationfor vector-valued lp-spaces. Therefore we will use an interpolation methodof Yoshikawa [12], where such spaces are directly involved.For 1 ≤ p <∞, ξ ∈ Rn and a Banach space X let lp(ξ,X) be the Banachspace of all families w = (w(v))v∈Zn such that w(v) ∈ X for every v ∈ Znand‖w‖lp(ξ,X) =(∑v∈Zn(2−ξ·v ‖w(v)‖X)p)1/pis finite. Let us notice that(2) ‖a‖θp;Kp = inf(n∑k=0‖ak‖plp(ξk,Ak))1/p,where the infimum is taken over all decompositions a =∑nk=0 ak(v) forevery v ∈ Zn with ak = (ak(v))v∈Zn ∈ lp(ξk, Ak) and k = 0, . . . , n.Particularly useful in our considerations will be another equivalent normintroduced by Yoshikawa (see [12, Définition 1.6]). For a ∈ Σ( ~A), θ ∈ Hn+1+and 1 ≤ p <∞ let(3) ‖a‖θp = inf{max0≤k≤n‖ak‖lp(ξk,Ak)},56 A. Kryczka and K. Kurlejwhere the infimum is taken as for ‖a‖θp;Kp . Since norms (2) and (3) areequivalent, the spaceAθp ={a ∈ Σ( ~A) : ‖a‖θp <∞}is equal to the Sparr space Aθp;K up to an equivalent norm (see [11, Remark4.6]).3. Results. Let [cij ] be an infinite matrix of scalars. A sequence (xj) ina Banach space X is called summable with respect to [cij ] if∑∞j=0 cijxjis convergent for each i ∈ N and the sequence (yi) with yi =∑∞j=0 cijxjconverges in X. The matrix [cij ] is called a regular method of summabilityif every sequence (xj) converging to x ∈ X is summable to x with respectto [cij ].We say that X has the Banach–Saks property with respect to the regularmethod of summability [cij ] (in short, the BS property with respect to [cij ]),if every bounded sequence in X has a subsequence summable with respectto [cij ].If cij ≥ 0 for all i, j, a regular method of summability [cij ] is calledpositive. This class of summability methods includes many classical meth-ods, such as the Hölder method (H, k) of order k ≥ 1, the regular Nørlundmethod (N, pn) and the Cesàro method (C, k) with k ≥ 1 (see [2], [5] fordetails).The Silverman–Toeplitz theorem [3] states that a matrix [cij ] is a regularmethod of summability if and only if:(i) there is M > 0 such that∑∞j=0 |cij | < M for every row i;(ii) limi→∞ cij = 0 for every column j;(iii) limi→∞∑∞j=0 cij = 1.The key result which we use in the proof of our theorem depends on thefollowing lemma proved in [6] with the use of the Galvin–Prikry theorem[4].Lemma 3.1. Let X be a Banach space with the Banach–Saks property withrespect to the regular method of summability [cij ]. Let (xj) be a boundedsequence in X. For every ε > 0 there exist N ≥ 0 and a subsequence (x′j)of (xj) such that for every subsequence (x′′j ) of (x′j) and all m,n ≥ N ,∥∥∥∥∥∥∞∑j=0cmjx′′j −∞∑j=0cnjx′′j∥∥∥∥∥∥ < ε.Let Y be a Banach space with a basis (ev) such that if |αv| ≤ |βv| for allv ∈ N, then ∥∥∥∥∥∞∑v=0αvev∥∥∥∥∥Y≤∥∥∥∥∥∞∑v=0βvev∥∥∥∥∥Y.Regular matrix methods of summability... 57In particular, every unconditional basis of a real Banach space with uncon-ditional constant 1 has this property (see [8]).Given a sequence (Xv) of Banach spaces, by a direct sum Y (Xv) we meanthe space of all sequences x = (x(v)) such that x(v) ∈ Xv for every v ∈ Nand∑∞v=0 ‖x(v)‖Xv ev ∈ Y . Then Y (Xv) is a Banach space with the norm‖x‖Y (Xv) =∥∥∥∥∥∞∑v=0‖x(v)‖Xv ev∥∥∥∥∥Y.The Cauchy criterion provided by Lemma 3.1 leads to the following generalresult proved in [6]. In particular, this result can be applied to spaceslp(ξ,X) used in the discrete interpolation methods.Theorem 3.2. A direct sum Y (Xv) has the Banach–Saks property withrespect to the regular positive method of summability [cij ] if and only if Yand all Xv have this property.Our aim is to prove that the BS property with respect to [cij ] is carriedover to interpolation spaces by even one space forming the interpolation fam-ily and possessing this property. The direct use of norm (3) is not enoughto obtain such result. On the other hand, as noticed in [7], the conclusionof Proposition 2.6 in [12] (that norm (3) can be expressed by a logarithmi-cally convex equality) is not justified. Nevertheless, using standard methodsapplied by Lions and Peetre in [9] for Banach couples, we can obtain forBanach families a logarithmically convex estimate up to a constant factor.There is a constant cθ > 0 such that for every a ∈ Aθp,‖a‖θp ≤ cθn∏k=0‖ak‖θklp(ξk,Ak),where a =∑nk=0 ak as in the definition of ‖a‖θp (see [7]).Theorem 3.3. Let [cij ] be a regular positive method of summability, θ ∈Hn+1+ and 1 < p <∞. If lp and at least one of the spaces A0, . . . , An havethe Banach–Saks property with respect to [cij ], then so have all real interpo-lation spaces Aθp with respect to the Banach (n+1)-tuple ~A = (A0, . . . , An).Proof. We show that if A0 has the BS property with respect to [cij ], theneach bounded sequence (aj) ⊂ Aθp contains a subsequence (a′j) such that(∑∞j=0 cija′j) is a Cauchy sequence. The proof for Ak with 1 ≤ k ≤ n runssimilarly.We restrict our attention to sequences (aj) in the open unit ball of Aθp.Then, for every j = 0, 1, 2, . . ., in the unit balls oflp(ξ0, A0), . . . , lp(ξn, An)there exist, respectively,x0j = (a0j(v))v∈Zn , . . . , xnj = (anj(v))v∈Zn58 A. Kryczka and K. Kurlejsuch thata0j(v) + . . .+ anj(v) = ajfor all v ∈ Zn.Fix ε > 0 and let θ = (θ0, . . . , θn) ∈ Hn+1+ . By the Silverman–Toeplitztheorem, there is M > 0 such that∑∞j=0 |cij | < M for every i ∈ N. Takeε0 > 0 such thatcθεθ00 (2M)1−θ0 < ε.Since lp and A0 have the BS property with respect to [cij ], by Theorem 3.2,lp(ξ0, A0) also has this property. Thus, for such chosen ε0, there exist N ≥ 0and a subsequence (x′0j) of (x0j) such that for all m,n ≥ N ,∥∥∥∥∥∥∞∑j=0cmjx′0j −∞∑j=0cnjx′0j∥∥∥∥∥∥lp(ξ0,A0)< ε0.For each k = 1, . . . , n let the subsequence (x′kj) of (xkj) correspond tothe subsequence (x′0j) of (x0j), that is x′kj = (a′kj(v))v∈Zn witha′0j(v) + . . .+ a′nj(v) = a′jfor all v ∈ Zn. It follows that for all m,n ≥ N ,∥∥∥∥∥∥∞∑j=0cmja′j −∞∑j=0cnja′j∥∥∥∥∥∥θp≤ cθn∏k=0∥∥∥∥∥∥∞∑j=0cmjx′kj −∞∑j=0cnjx′kj∥∥∥∥∥∥θklp(ξk,Ak)= cθ∥∥∥∥∥∥∞∑j=0cmjx′0j −∞∑j=0cnjx′0j∥∥∥∥∥∥θ0lp(ξ0,A0)n∏k=1∥∥∥∥∥∥∞∑j=0cmjx′kj −∞∑j=0cnjx′kj∥∥∥∥∥∥θklp(ξk,Ak)≤ cθεθ00n∏k=1 ∞∑j=0|cmj − cnj |∥∥x′kj∥∥lp(ξk,Ak)θk≤ cθεθ00n∏k=1 ∞∑j=0|cmj |+∞∑j=0|cnj |maxj≥0∥∥x′kj∥∥lp(ξk,Ak)θk≤ cθεθ00n∏k=1(2M)θk= cθεθ00 (2M)1−θ0 < ε.Regular matrix methods of summability... 59Thus (∑∞j=1 cija′j) is a Cauchy sequence and Aθp has the BS property withrespect to [cij ]. The K-method of Sparr [11] is a generalization of the Lions–Peetre [9]real interpolation method defined for Banach couples. For n = 1 the vectorθ = (θ0, θ1) ∈ H2+ is replaced by a parameter θ ∈ (0, 1) which corresponds to(1 − θ, θ) in Sparr’s notation. The following corollary extends Beauzamy’sresult [1] on the interpolation of the classical Banach–Saks property to allpositive regular methods of summability.Corollary 3.4. If lp and at least one of the spaces A0, A1 have the Banach–Saks property with respect to [cij ], then so have all Lions–Peetre’s real in-terpolation spaces Aθp with respect to the interpolation couple ~A = (A0, A1).References[1] Beauzamy, B., Propriété de Banach–Saks, Studia Math. 66 (1979/80), 227–235.[2] Boos, J., Classical and Modern Methods in Summability. Assisted by Peter Cass,Oxford University Press, Oxford, 2000.[3] DeVito, C. L., Functional Analysis, Academic Press, Inc., New York–London, 1978.[4] Galvin, F., Prikry, K., Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973),193–198.[5] Hardy, G. H., Divergent Series, the Clarendon Press, Oxford, 1949.[6] Kryczka, A., Kurlej, K., Regular methods of summability for direct sums of Banachspaces, J. Math. Anal. Appl. 494 no. 1, (2021), 124636.[7] Kutzarova, D., Nikolova, L. I., Prus, S., Infinite-dimensional geometric properties ofreal interpolation spaces, Math. Nachr. 191 (1998), 215–228.[8] Lindenstrauss, J., Tzafriri, L., Classical Banach Spaces I, Springer-Verlag, New York,1977.[9] Lions, J.-L., Peetre, J., Sur une classe d’espaces d’interpolation, Inst. Hautes ÉtudesSci. Publ. Math. 19 (1964), 5–68.[10] Pietsch, A., History of Banach Spaces and Linear Operators, Birkhäuser, Boston,2007.[11] Spar, G., Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99 (1974),247–316.[12] Yoshikawa, A., Sur la théorie d’espaces d’interpolation—les espaces de moyenne deplusieurs espaces de Banach, J. Fac. Sci. Univ. Tokyo 16 (1970), 407–468.Andrzej Kryczka Konrad KurlejInstitute of Mathematics LublinMaria Curie-Skłodowska University Poland20-031 Lublin e-mail: konrad.kurlej@gmail.comPolande-mail: andrzej.kryczka@umcs.plReceived April 26, 2021

Andrzej Kryczka

Konrad Kurlej

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Regular matrix methods of summability and real interpolation

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University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

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Regular matrix methods of summability and real interpolation

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University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

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Biblioteka Nauki - repozytorium artykuÅÃ³w