@ Appl. Gen. Topol. 22, no. 2 (2021), 367-383doi:10.4995/agt.2021.15061 © AGT, UPV, 2021 Lipschitz integral operators represented by vector measures Elhadj Dahia a and Khaled Hamidi b a Laboratoire de Mathématiques et Physique Appliquées, École Normale Supérieure de Bousaada, 28001 Bousaada, Algeria (hajdahia@gmail.com) b Department of Mathematics, University of Mohamed El-Bachir El-Ibrahimi, Bordj Bou Arréridj, 34030 El-Anasser, Algeria, and Laboratoire d’Analyse Fonctionnelle et Géométrie des Espaces, University of M’sila, 28000 M’sila, Algeria. (khaled.hamidi@univ-bba.dz) Communicated by S. Romaguera Abstract In this paper we introduce the concept of Lipschitz Pietsch-p-integral mappings, (1 ≤ p ≤ ∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vector measure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), Lp(µ, K) and L∞(µ, K). Also we show that this type of opera- tors fits in the theory of composition Banach Lipschitz operator ideals. For p = ∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given. 2010 MSC: 47B10; 47L20; 26A16. Keywords: Lipschitz Pietsch-p-integral operators; Lipschitz strictly p- integral operators; vector measure representation. Introduction The class of p-integral linear operators was introduced in 1969 by Persson and Pietsch [18] (also known as strictly p-integral or Pietsch-p-integral op- erators) establishing many of its fundamental properties using the theory of Received 02 February 2021 – Accepted 13 May 2021 http://dx.doi.org/10.4995/agt.2021.15061 E. Dahia and K. Hamidi vector measures. In 1989, Cardassi studied the factorization properties and some results of coincidences for these operators in [6]. The ideal of p-integral polynomials on Banach spaces has been defined and characterized by Cilia and Gutiérrez, in [9] for p = 1 and in [8] for p ≥ 1, as a natural polynomial extension of Pietsch-p-integral operators. In this paper we introduce and study the Lipschitz version of this concept. We define the Lipschitz Pietsch-p-integral operator (1 ≤ p ≤∞) as a Lipschitz mapping between a pointed metric space and a Banach space by an integral representation with respect to a vector measure on the Borel σ-algebra of a compact Hausdorff space K. Special attention is paid to the factorization of these mappings and we compare our class with some well known Lipschitz operators defined by a factorization schema or by summability of series. Note that the class of Lipschitz Pietsch-1-integral operators is studied in [5]. In this case, the authors use only factorization schemes to define this concept without using vector measure theory. We describe now the contents of the present paper. After this introduction, in section one we fix notation and basic concepts related to Lipschitz mappings and vector measure of interest for our purposes. In section two we extend to Lipschitz mappings the concept of Pietsch-p-integral operators for p ≥ 1 and we prove a factorization theorem for these mappings through the classical Ba- nach spaces C(K) and Lp(µ, K). The third section is devoted to study the notion of Lipschitz Pietsch-∞-integral operators, starting from the representa- tion by a vector measure, we present a characterization given by a factorization through a linear weakly compact operator. Finally, in section four we establish the relationship between our class and the class of Lipschitz p-summing opera- tors, Lipschitz Grothendieck-p-integral operators, strongly Lipschitz p-nuclear operators and Lipschitz weakly compact operators. 1. Notation and preliminaries The notation used in the paper is in general standard. E and F are real Banach spaces. X, Y and Z will be pointed metric spaces with a base point denoted by 0 and a metric denoted by d. Given a Banach space E, E∗ denotes its topological dual, and BE its closed unit ball. As usual, L(E, F) denotes the space of all continuous linear operators from E to F with the operator norm. A Banach space E will be considered as a pointed metric space with distinguished point 0 and distance d(x, y) = ‖x−y‖. With Lip0(X, E) we denote the Banach space of all Lipschitz mappings from X to E, taking 0 into 0, under the Lipschitz norm Lip(T ) = inf {C > 0 : ‖T (x) −T (x′)‖≤ Cd(x, x′)} . Moreover, T is called an isometric embedding if ‖T (x) −T (x′)‖ = d(x, x′) for all x, x′ ∈ X. When E = R, Lip0(X, R) is denoted by X # and it is called the Lipschitz dual of X. Along the paper we consider BX# endowed with the pointwise topology (BX# is a compact Hausdorff space in this topology). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 368 Lipschitz integral operators represented by vector measures It is well know that X# has a predual, namely the space of Arens and Eells Æ(X) of the metric space X [2] (also known as the Lipschitz-free Banach space F(X) of X [14]). This space is one of the main tools that we will use in the sequel. We summarize some basic properties of Æ(X). A molecule on X is a scalar valued function m on X with finite support that satisfies ∑ x∈X m (x) = 0. We denote by M(X) the linear space of all molecules on X. For x, x′ ∈ X the molecule mxx′ is defined by mxx′ = χ{x}−χ{x′}, where χA is the characteristic function of the set A. For m ∈ M(X) we can write m = n∑ i=1 λimxix′i for some suitable scalars λi, and we write ‖m‖M(X) = inf { n∑ i=1 |λi|d(xi, x ′ i), m = n∑ i=1 λimxix′i } , where the infimum is taken over all representations of the molecule m. Denote by Æ(X) the completion of the normed space (M(X),‖.‖M(X)). The map kX : X →Æ(X) defined by kX(x) = mx0 isometrically embeds X in Æ(X). For any T ∈ Lip0(X, E) there exists a unique linear map TL ∈ L(Æ (X) , E) such that T = TL ◦kX, i.e. the following diagram commutes X T // kX �� E Æ (X) . TL ;; ① ① ① ① ① ① ① ① ① Moreover, ‖TL‖ = Lip(T ) (see [19, Theorem 2.2.4 (b)]). The operator TL is referred to as the linearization of T . The correspondence T ←→ TL establishes an isometric isomorphism between the Banach spaces Lip0(X, E) and L(Æ (X) , E). In particular, the spaces X# and Æ(X)∗ are isometrically isomorphic via the linearization R(f) := fL, where fL(m) = ∑ x∈X f(x)m(x), in particular fL(mxx′) = f(x) −f(x ′), (see [19, Theorem 2.2.2]). Now we recall some simple notions from vector measure theory. Let (Ω, Σ) be a measurable space and E a Banach space. Let m : Σ −→ E be a countably additive vector measure [11, Definition I.1.1]. For every x∗ ∈ E∗, let 〈m, x∗〉 be the scalar signed measure defined by 〈m, x∗〉(A) := 〈m(A), x∗〉 for all A ∈ Σ. The semivariation of m is the subadditive real bounded set function ‖m‖ : A ∈ Σ −→‖m‖(A) ∈ [0, +∞) defined by ‖m‖(A) = sup{|〈m, x∗〉|(A) : ‖x∗‖≤ 1} , where |〈m, x∗〉| is the variation measure of the signed measure 〈m, x∗〉. According to [17, Page 106] and [16, Definition 2.1], a measurable (scalar valued) function f is integrable with respect to m if • It is integrable with respect to the scalar measure 〈m, x∗〉 for every x∗ ∈ E∗. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 369 E. Dahia and K. Hamidi • For every A ∈ Σ there exists an element mf (A) ∈ E satisfying 〈mf (A), x ∗〉 = ∫ A fd〈m, x∗〉 , x∗ ∈ E∗. We will use the classical notation mf (A) := ∫ A fdm, A ∈ Σ. For the general theory of vector measures we refer the reader to the classical monograph [11]. For 1 ≤ p ≤ ∞, linear Pietsch-p-integral operators were introduced by Persson and Pietsch [18] and deeply studied in [6, 10] among others. Definition 1.1. The linear operator u : E −→ F , between Banach spaces, is Pietsch-p-integral (1 ≤ p < ∞) if there are a regular Borel countably additive vector measure m of bounded semivariation on B(BE∗), (where B(BE∗) is the Borel σ-algebra of BE∗), and a positive regular Borel measure µ on BE∗ such that u (x) = ∫ BE∗ 〈x, x∗〉dm(x∗), x ∈ E, and ∥∥∥∥ ∫ BE∗ fdm ∥∥∥∥ ≤ (∫ BE∗ |f| p dµ ) 1 p , f ∈ C(BE∗). The Banach space of these operators is denoted by PIp(E, F) under the norm defined by ‖u‖PIp = inf µ(BE∗) 1 p , where the infimum is taken over all measures µ satisfying the above inequality. For p = ∞, u is called Pietsch-∞-integral if there is a regular Borel countably additive vector measure m : B(BE∗) −→ F of bounded semivariation such that u (x) = ∫ BE∗ 〈x, x∗〉dm(x∗), x ∈ E. In this case, ‖T‖PIL ∞ = inf ‖m‖(BE∗), taking the infimum over all m that satisfy the above equality. In [18, Satz 15 and Satz 17] we find some canonical linear Pietsch-p-integral operators that will be used in the sequel. Let K be a compact Hausdorff space and ν be a positive regular Borel measure on K. Let (Ω, Σ, µ) be a finite measure space and jp, ip be the inclusions of C(K) into Lp(K, ν) and of L∞(µ) into Lp(µ) respectively for 1 ≤ p < ∞. Then jp ∈ PIp(C(K), Lp(K, ν)) and ip ∈PIp(L∞(µ), Lp(µ)) with ‖jp‖PIp = ‖jp‖ = ν(K) 1 p and ‖ip‖PIp = ‖ip‖ = µ(Ω) 1 p . Remark 1.2. Note that by using [8, Theorem 2.5], u ∈ PIp (E, F) if and only if there are a compact Hausdorff space K, an embedding h : E −→ C(K), a regular Borel countably additive vector measure m : B(K) −→ F of bounded © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 370 Lipschitz integral operators represented by vector measures semivariation and a positive regular Borel measure µ on K such that for all x ∈ E, u(x) = ∫ K h (x) dm, and ∥∥∥∥ ∫ K fdm ∥∥∥∥ ≤ (∫ K |f| p dµ ) 1 p for all f ∈ C (K). In this case ‖u‖PIp = inf ‖h‖µ(K) 1 p , where the infimum is taken over all K, m and h as above. 2. Lipschitz Pietsch-p-integral operators Definition 2.1. Let X be a pointed metric space, E a Banach space and let T ∈ Lip0(X, E). For 1 ≤ p < ∞, the mapping T is said to be Lipschitz Pietsch-p-integral operator if there are a regular Borel probability measure space (Ω, Σ, µ), a linear operator A ∈ L(Lp(µ), E) and a Lipschitz operator B ∈ Lip0(X, L∞(µ)) giving rise to the following commutative diagram (2.1) X B �� T // E L∞(µ) ip // Lp(µ). A OO where ip : L∞(µ) −→ Lp(µ) is the canonical mapping. The set of all Lipschitz Pietsch-p-integral mappings from X to E is denoted by PILp (X, E). With each T ∈ PILp (X, E) we associate its Lipschitz Pietsch-p-integral quantity, ‖T‖PILp = inf ‖A‖Lip(B), where the infimum is taken over all µ, A and B as above. Remark 2.2. (1) As an easy consequence of the definition, if T ∈ PILp (X, E) we have Lip(T ) ≤‖T‖PIL p . (2) Notice that the definition is the same if we consider a finite regular Borel measure space (Ω, Σ, µ), in this case, for T ∈ PILP (X, E) we have ‖T‖PIL P = inf ‖A‖µ(Ω) 1 p Lip(B), where the infimum is taken over all µ, A and B in (2.1 ). (3) We don’t know if being Lipschitz Pietsch-p-integrability implies Pietsch- p-integrability whenever the mapping T is linear. The converse is of course clearly true, that is if E and F are Banach spaces and T : E −→ F is linear Pietsch-p-integral then T is Lipschitz Pietsch-p- integral and ‖T‖PILp ≤‖T‖PIp. We have the following immediate consequence of the definition above. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 371 E. Dahia and K. Hamidi Proposition 2.3 (Inclusion Theorem). Let 1 ≤ p ≤ q < ∞. Then PILp (X, E) ⊂ PILq (X, E) and ‖T‖PIL q ≤‖T‖PIL p for all T ∈PILp (X, E). In order to prove the factorization theorem for the class of Lipschitz Pietsch- p-integral operators, (1 ≤ p ≤∞) we need the following technical lemma. Lemma 2.4. Let J : M(X) −→ C(BX#) be the operator defined by J(m)(f) = n∑ i=1 λi(f(xi) −f(x ′ i)), for all m = ∑n i=1 λimxix′i ∈ M(X) and f ∈ BX#. Then this operator is an isometric embedding. Proof. Since Æ(X)∗ and X# are isometrically isomorphic via the linearization, for all f ∈ X# there is m∗ ∈ Æ (X) ∗ such that fL = m ∗. For all m ∈ M(X) we have ‖J(m)‖ C(B X# ) = sup f∈B X# |J(m)(f)| = sup ‖fL‖≤1 ∣∣∣∣∣ n∑ i=1 λi(fL(mxix′i) ∣∣∣∣∣ = sup ‖m∗‖≤1 |〈m, m∗〉| = ‖m‖Æ(X) = ‖m‖M(X) , and the proof follows. � For x ∈ X, we denote by δx the functional δx : X # −→ R defined as δx(f) = f(x), f ∈ X #. Let ιX : X −→ C(BX#) the natural Lipschitz isometric embedding such that ιX(x) is the restriction of δx to BX#, for all x ∈ X. The following theorem gives a parallel development of the factorization schemes concerning Lipschitz Pietsch-p-integral operators that highlights the role of the space C (BX#). Theorem 2.5. Let 1 ≤ p < ∞ and let T ∈ Lip0(X, E). Then T is Lipschitz Pietsch-p-integral if and only if there exist a regular Borel probability measure ν on BX# and an operator à ∈ L(Lp(ν), E) such that the following diagram commutes (2.2) X ιX �� T // E C (BX#) jp // Lp(ν) à OO , where jp is the canonical map. Moreover, ‖T‖PILp = inf {∥∥∥à ∥∥∥ : T = Ã◦ jp ◦ ιX } . Proof. We write △ for the proposed infimum. Suppose that T admits a fac- torization (2.2). If j∞ is the canonical inclusion map from C (BX#) to L∞(ν), we have the factorization T = Ã◦ ip ◦ j∞ ◦ ιX : X ιX −→ C (BX#) j∞ −→ L∞(ν) ip −→ Lp(ν) à −→ E. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 372 Lipschitz integral operators represented by vector measures Denoting by B = j∞◦ιX, it follows that B ∈ Lip0(X, L∞(ν)) and Lip(B) ≤ 1, which implies that T is Lipschitz Pietsch-p-integral and ‖T‖PILp ≤ ∥∥∥à ∥∥∥ Lip(B) ≤ ∥∥∥à ∥∥∥ . Passing to the infimum we get ‖T‖PIL p ≤△. Conversely, suppose that T ∈ PILp (X, E). Fix ε > 0, there are a regular Borel probability measure space (Ω, Σ, µ), an operator A ∈L(Lp(µ), E) and a Lipschitz mapping B ∈ Lip0(X, L∞(µ)) such that T = A◦ ip ◦B : X B −→ L∞(µ) ip −→ Lp(µ) A −→ E, and ‖A‖Lip(B) ≤‖T‖PIL p +ε. Let BL ∈L(Æ(X), L∞(µ)) be the linearization of the Lipschitz mapping B, that is B = BL◦kX and ‖BL‖ = Lip(B). Consider the natural extension of the isometric embedding J, mentioned in Lemma 2.4, to Æ(X) which we denote also by J. The injectivity of L∞(µ) assures the existence of an operator B̃L ∈L(C(BX#), L∞(µ)) that extends BL with ∥∥∥B̃L ∥∥∥ = ‖BL‖, that is BL = B̃L ◦J or the following diagram commutes Æ (X) BL // J �� L∞(µ) C(BX# ) B̃L 99 s s s s s s s s s . The operator ip : L∞(µ) −→ Lp(µ) is p -summing with p-summing norm one then ip ◦ B̃L is too with πp(ip ◦ B̃L) ≤ ∥∥∥B̃L ∥∥∥. By [10, Corollary 2.15] there exist a regular Borel probability measure ν on BX# and an operator S ∈L(Lp(ν), Lp(µ)) such that ip ◦ B̃L = S ◦ jp : C (BX#) jp −→ Lp(ν) S −→ Lp(µ), and πp(ip ◦ B̃L) = ‖S‖ . Then T = (A ◦S)◦ jp ◦ (J ◦kX) : X J◦kX −→ C (BX#) jp −→ Lp(ν) A◦S −→ E. Easy calculations prove that J ◦ kX = ιX, which implies that T admits a factorization of the form (2.2) with à = A ◦S and we have △ ≤ ∥∥∥à ∥∥∥ ≤‖A‖πp(ip ◦ B̃L) ≤ ‖A‖ ∥∥∥B̃L ∥∥∥ = ‖A‖Lip(B) ≤‖T‖PIL p + ε. Since this holds for all ε > 0 we arrive at △≤‖T‖PIL p . � The next theorem is the main result of this section and provides a character- ization of the class of Lipschitz Pietsch-p-integral operators, that is an integral representation with respect to a vector measure. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 373 E. Dahia and K. Hamidi Theorem 2.6. Let 1 ≤ p < ∞ and let T ∈ Lip0(X, E). Then T is Lips- chitz Pietsch-p-integral if and only if there are a compact Hausdorff space K, a Lipschitz embedding φ : X −→ C(K) with φ(0) = 0, a regular Borel count- ably additive vector measure m : B(K) −→ E of bounded semivariation and a positive regular Borel measure µ on K such that (2.3) T (x) = ∫ K φ (x) dm, x ∈ X, and (2.4) ∥∥∥∥ ∫ K fdm ∥∥∥∥ ≤ (∫ K |f| p dµ ) 1 p , for all f ∈ C (K) . In this case ‖T‖PILp = inf { Lip(φ)µ (K) 1 p } , where the infimum is taken over all K, φ, m and µ satisfying (2.3) and (2.4). Proof. Suppose that T ∈PILp (X, E), and fix ε > 0. There are a regular Borel probability measure ν on BX# and à ∈L(Lp (ν) , E) such that T = à ◦ jp ◦ ιX : X ιX −→ C (BX#) jp −→ Lp(ν) à −→ E, and ∥∥∥à ∥∥∥ ≤‖T‖PIL p +ε. The linear operator Ã◦jp : C (BX#) −→ E is Pietsch- p-integral with ∥∥∥à ◦ jp ∥∥∥ PIp ≤ ∥∥∥à ∥∥∥‖jp‖PIp ≤ ∥∥∥à ∥∥∥ ≤‖T‖PILp + ε. By Remark 1.2, there are a compact Hausdorff space K, an embedding h : C (BX#) −→ C(K), a regular Borel countably additive vector measure m : B(K) −→ E of bounded semivariation and a positive regular Borel measure µ on K such that for all x ∈ X, T (x) = à ◦ jp (ιX(x)) = ∫ K h (ιX(x)) dm, ∥∥∥∥ ∫ K fdm ∥∥∥∥ ≤ (∫ K |f| p dµ ) 1 p for all f ∈ C (K) and ‖h‖µ (K) 1 p ≤ ∥∥∥Ã◦ jp ∥∥∥ PIp + ε. Which means that (2.3) and (2.4) are true by taking into account that φ = h ◦ ιX is a Lipschitz embedding from X to C(K) vanishing at 0 with Lip(φ) ≤‖h‖ . Moreover Lip(φ)µ (K) 1 p ≤ ∥∥∥à ◦ jp ∥∥∥ PIp + ε ≤‖T‖PILp + 2ε. Since this holds for every ε > 0, it follows that Lip(φ)µ (K) 1 p ≤‖T‖PILp . Conversely, suppose that T satisfies the conditions (2.3) and ( 2.4). By [11, Theorem VI.2.1] there exists u ∈ L(C (K) , E) such that u(f) = ∫ K fdm, f ∈ C (K) . Consider the canonical mapping jp = ip ◦ j∞ : C (K) j∞ −→ L∞ (K, µ) ip −→ Lp (K, µ) , © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 374 Lipschitz integral operators represented by vector measures and define R : jp (C (K)) −→ E by R (jp (f)) := u(f). The linear mapping R is well-defined and continuous with norm ≤ 1 since for all f ∈ C (K) , ‖R (jp (f))‖ = ∥∥∥∥ ∫ K fdm ∥∥∥∥ ≤ (∫ K |f| p dµ ) 1 p = ‖jp (f)‖ . By [12, Lemma IV.8.19] we have jp (C (K)) = Lp (K, µ) , so R can be extended to a continuous linear operator R̃ : Lp (K, µ) −→ E with ∥∥∥R̃ ∥∥∥ ≤ 1. If we put B = j∞ ◦ φ, we obtain B ∈ Lip0(X, L∞ (K, µ)) and Lip(B) ≤ Lip(φ). On the other hand, R̃ ◦ ip ◦B(x) = R̃ ◦ jp ◦φ(x) = u(φ(x)) = ∫ K φ(x)dm = T (x), and therefore T factors as in (2.1), that is T ∈PILp (X, E) with ‖T‖PILp ≤ Lip(B) ∥∥∥R̃ ∥∥∥ µ (K) 1 p ≤ Lip(φ)µ (K) 1 p . � Now we present a relationship between the Lipschitz Pietsch-p-integral op- erator and its linearization. Theorem 2.7. Let T ∈ Lip0(X, E) and 1 ≤ p < ∞. Then T ∈ PI L p (X, E) if and only if TL ∈PIp(Æ (X) , E). Moreover, we have (2.5) ‖T‖PILp = ‖TL‖PIp . Proof. Suppose that TL ∈PIp(Æ (X) , E). According to [18, Satz 18], for every ε > 0 we can choose a typical factorization of TL TL = A ◦ ip ◦B : Æ (X) B −→ L∞ (µ) ip −→ Lp (µ) A −→ E, such that A ∈ L(Lp (µ) , E) and B ∈ L(Æ (X) , L∞ (µ)) with ‖A‖‖B‖ ≤ ‖TL‖PIp+ε. It is clear that the mapping R := B◦kX belongs to Lip0 (X, L∞ (µ)) and Lip(R) ≤ ‖B‖. The factorization T = TL ◦ kX = A ◦ ip ◦ R implies that T ∈PILp (X, E) and ‖T‖PIL p ≤‖A‖Lip (R) ≤‖TL‖PIp + ε. Conversely, if T ∈ PILp (X, E), for ε > 0 choose the following factorization of T T = A◦ ip ◦B : X B −→ L∞ (µ) ip −→ Lp (µ) A −→ E, such that A ∈ L(Lp (µ) , E) and B ∈ Lip0(X, L∞ (µ)) with ‖A‖Lip(B) ≤ ‖T‖PILp + ε. The uniqueness of the linearization maps gives that TL = (A ◦ ip ◦B)L = A◦ ip ◦BL. Then, we have that TL ∈PIp(Æ (X) , E) with ‖TL‖PIp ≤‖A‖‖BL‖ = ‖A‖Lip(B) ≤‖T‖PILp + ε. The proof concludes. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 375 E. Dahia and K. Hamidi The notion of Lipschitz operator ideal was introduced by Achour, Rueda, Sánchez-Pérez and Yahi [1]. This can be seen as an extension of the notion of linear Banach operator ideal. A Lipschitz operator ideal ILip is a subclass of Lip0 such that for every pointed metric space X and every Banach space E the components ILip(X, E) := Lip0(X, E)∩ILip satisfy (i) ILip(X, E) is a linear subspace of Lip0(X, E). (ii) vg ∈ILip(X, E) for v ∈ E and g ∈ X #. (iii) The ideal property: if S ∈ Lip0(Y, X), T ∈ ILip(X, E) and w ∈ L(E, F), then the composition wT S is in ILip(Y, F). A Lipschitz operator ideal ILip is a normed (Banach) Lipschitz operator ideal if there is a function ‖.‖ILip : ILip −→ [0, +∞[ that satisfies (i’) For every pointed metric space X and every Banach space E, the pair (ILip(X, E),‖.‖ILip) is a normed (Banach) space and Lip(T ) ≤‖T‖ILip for all T ∈ILip(X, E). (ii’) ‖IdK : K −→ K, IdK(λ) = λ‖ILip = 1. (iii’) If S ∈ Lip0(Y, X), T ∈ILip(X, E) and w ∈L(E, F), then ‖wT S‖ILip ≤ Lip(S)‖T‖ILip ‖w‖ . Following [1, Definition 3.1], there is a way to construct a (Banach) Lips- chitz operator ideal from a (Banach) linear operator ideal, called composition method. Let A be a (Banach) linear operator ideal. A Lipschitz mapping T ∈ Lip0(X, E) belongs to the composition Lipschitz operator ideal A◦ Lip0 if there exists a Banach space F, a Lipschitz operator S ∈ Lip0(X, F) and a linear operator u ∈ A(F, E) such that T = u ◦ S. If (A,‖.‖A) is a Banach operator ideal we write ‖T‖A◦Lip0 = inf ‖u‖A Lip(S), where the infimum is taken over all u and S as above. In [1], the authors establish a criterion to decide whenever a Lipschitz oper- ator ideal is of composition or not. Proposition 2.8 ([1, Proposition 3.2]). Let X be a pointed metric space, E a Banach space and A an operator ideal. A Lipschitz operator T ∈ Lip0(X, E) be- longs to A◦Lip0(X, E) if and only if its linearization TL belongs to A(F(X), E). Furthermore, if (A,‖·‖A) is a Banach operator ideal then (A◦Lip0,‖·‖A◦Lip0) is Banach Lipschitz operator ideal with ‖T‖A◦Lip0 = ‖TL‖A . By Theorem 2.7 and the above criterion, we have the following. Proposition 2.9. ( PILp ,‖·‖PIL p ) is the Banach Lipschitz operator ideal gen- erated by the composition method from the Banach operator ideal PIp. In other words PILp (X, E) = PIp ◦Lip0(X, E) isometrically © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 376 Lipschitz integral operators represented by vector measures for every pointed metric space X and every Banach space E. We say that a pointed metric space W is 1-injective (or an absolute Lipschitz retract) if for every metric space X, every subset X0 of X and every Lipschitz mapping T ∈ Lip0(X0, W) there is a Lipschitz mapping T̃ ∈ Lip0(X, W) ex- tending T with Lip(T ) = Lip(T̃). The real Banach space L∞(µ) for a finite measure µ is 1-injective (see [4, Chapter 1]). By the typical Pietsch-p-integral factorization of a Lipschitz mapping T , we can find a Pietsch-p-integral extension T̃ . Proposition 2.10. Let X and Z be pointed metric spaces with X ⊂ Z and let E be a Banach space. Each Lipschitz Pietsch-p-integral operator T : X −→ E admits a Lipschitz Pietsch-p-integral extension T̃ : Z −→ E with ‖T‖PIL p = ∥∥∥T̃ ∥∥∥ PIL p . Proof. If T ∈PILp (X, E), then for all ε > 0 there are a regular Borel probability measure space (Ω, Σ, µ), A ∈L(Lp(µ), E) and B ∈ Lip0(X, L∞(µ)) such that T = A◦ ip ◦B : X B −→ L∞(µ) ip −→ Lp(µ) A −→ E, and Lip(B)‖A‖ ≤ ‖T‖PIL p + ε. Since L∞(µ) is 1-injective, B admits an ex- tension B̃ ∈ Lip0(Z, L∞(µ)) with Lip(B̃) = Lip(B) i.e., the following diagram commutes X B // i �� L∞(µ) Z. B̃ ;; ① ① ① ① ① ① ① ① , where i ∈ Lip0(X, Z) is the natural isometric embedding. This creates a Lip- schitz Pietsch-p-integral extension T̃ : Z −→ E of T having the following factorization T̃ = A ◦ ip ◦ B̃ : Z B̃ −→ L∞(µ) ip −→ Lp(µ) A −→ E. Furthermore, ∥∥∥T̃ ∥∥∥ PILp ≤ Lip(B̃)‖A‖ = Lip(B)‖A‖≤‖T‖PILp + ε. Since this holds for all ε > 0 we get ∥∥∥T̃ ∥∥∥ PIL p ≤ ‖T‖PILp . For the reverse inequality, note that ‖T‖PIL p = ∥∥∥T̃ ◦ i ∥∥∥ PIL p ≤ ∥∥∥T̃ ∥∥∥ PIL p . � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 377 E. Dahia and K. Hamidi 3. Lipschitz Pietsch-∞-integral operators In this section we extend the definition of the class of Pietsch-∞-integral lin- ear operators to the case of Lipschitz operators and we will show a factorization theorem that characterizes these mappings. Definition 3.1. We say that a Lipschitz operator T ∈ Lip0(X, E) is Lipschitz Pietsch-∞-integral if there is a regular Borel countably additive vector measure m : B(BX#) −→ E of bounded semivariation such that (3.1) T (x) = ∫ B X# f (x) dm(f), x ∈ X. We denote by PIL∞(X, E) the set of all these mappings and we put ‖T‖PIL ∞ = inf ‖m‖(BX#), taking the infimum over all m such that (3.1) holds. Remark 3.2. If T ∈ PIL∞(X, E) then Lip(T ) ≤ ‖T‖PIL ∞ . In order to see this, for ε > 0 choose m such that ‖m‖(BX#) ≤‖T‖PIL ∞ + ε and for all x, y ∈ X, ‖T (x)−T (y)‖ ≤ ∫ B X# |f (x) −f (y)|dm(f) ≤ ‖m‖(BX#)d(x, y) ≤ (ε +‖T‖PIL ∞ )d(x, y). Hence, Lip(T ) ≤‖T‖PIL ∞ . Now we prove the main result of this section. We characterize the Pietsch- ∞-integral Lipschitz operators by means of a factorization scheme through a weakly compact linear operator. Theorem 3.3. For a Lipschitz operator T ∈ Lip0(X, E), the following state- ments are equivalent. (1) T is Lipschitz Pietsch-∞-integral. (2) There are a compact Hausdorff space K, a Lipschitz embedding ϕ ∈ Lip0(X, C(K)) and a weakly compact linear operator S ∈L(C(K), E) such that the following diagram commutes (3.2) X T // ϕ �� E C(K). S << ② ② ② ② ② ② ② ② (3) There are a regular Borel finite measure space (Ω, Σ, µ), a weakly com- pact operator R ∈L(L∞(µ), E) and a Lipschitz embedding φ ∈ Lip0(X, L∞(µ)) © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 378 Lipschitz integral operators represented by vector measures giving rise to the following commutative diagram (3.3) X T // φ �� E L∞(µ). R ;; ① ① ① ① ① ① ① ① ① In addition, ‖T‖PIL ∞ = inf ‖S‖Lip(ϕ) = inf ‖R‖Lip(φ). Where the first infimum is taken over all S and ϕ as in (3.2) and the second is taken over all R and φ as in (3.3). Proof. (1)=⇒(2). Take T ∈ PIL∞(X, E). For every ε > 0 choose m sat- isfying (3.1) and ‖m‖(BX#) ≤ ‖T‖ PIL ∞ + ε. Consider the linear operator S : C(BX# ) −→ E defined by S(h) = ∫ B X# hdm, for all h ∈ C(BX# ) and the natural Lipschitz isometric embedding ιX : X −→ C(BX#). In this case, for all x ∈ X we can write S ◦ ιX(x) = ∫ B X# ιX(x)(f)dm(f) = ∫ B X# f(x)dm(f) = T (x). Theorem VI.2.5 in [11] asserts that S is weakly compact with norm ‖S‖ = ‖m‖(BX#) and then ‖S‖Lip(ιX) = ‖m‖(BX#) ≤‖T‖PIL ∞ + ε. (2)=⇒(3). There is a a regular Borel countably additive vector measure m : B(K) −→ E of bounded semivariation such that S(f) = ∫ K fdm for all f ∈ C(K) and ‖m‖(K) = ‖S‖ (see [11, Theorem VI.2.1, VI.2.5 and Corollary VI.2.14]). It follows that T (x) = S ◦ϕ(x) = ∫ K ϕ(x)dm, x ∈ X. On the other hand, [11, Corollary I.2.6 and Theorem I.2.1] assures the existence of a regular Borel finite measure µ on B(K) such that m(A) = 0 for all A ∈ B(K) which satisfy that µ(A) = 0. Define the operator R ∈L(L∞(µ), E) by R(f) = ∫ K fdm, f ∈ L∞(µ) with ‖R‖ = ‖m‖(K) (see [11, Theorem I.1.13]). This operator is weakly compact (see [11, Definition I.1.14 and Theorem VI.1.1]). Consequently, R ◦ (j∞ ◦ϕ) = ∫ K j∞ ◦ϕdm = ∫ K φdm = T. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 379 E. Dahia and K. Hamidi (3)=⇒(1). As in the proof of the second implication of Theorem 2.5, starting from the diagram (3.3), consider the linearization φL of φ ∈ Lip0(X, L∞(µ)) and let φ̃L ∈ L(C(BX#), L∞(µ)) be the extension of φL, i.e., the following diagram commutes Æ (X) φL // J �� L∞(µ) C(BX# ). φ̃L 99 s s s s s s s s s s The linear operator R ◦ φ̃L : C(BX#) −→ E is weakly compact. Let m be the representing vector measure of R ◦ φ̃L, that is R ◦ φ̃L(f) = ∫ B X# fdm for all f ∈ C(BX#) and ‖m‖(BX#) = ∥∥∥R ◦ φ̃L ∥∥∥. It follows that T (x) = R ◦φ(x) = R ◦ φ̃L ◦J ◦kX(x) = ∫ B X# J ◦kX(x)dm, for all x ∈ X, and then T ∈PIL∞(X, E) and ‖T‖PIL ∞ ≤‖m‖(BX#) ≤‖R‖ ∥∥∥φ̃L ∥∥∥ = ‖R‖Lip(φ). Since this is true for every factorization as (3.3), we have‖T‖PIL ∞ ≤‖R‖Lip(φ). In order to show the reverse inequality, take T ∈ PIL∞(X, E) and ε > 0. Then there is m : B(BX#) −→ E (as in Definition 3.1) such that (3.1) is true and ‖m‖(BX#) ≤ ε + ‖T‖PIL ∞ . Following the proof of (2)=⇒(3), we can find a regular Borel finite measure µ on BX# and a weakly compact operator R ∈L(L∞(µ), E) represented by m such that ‖R‖Lip(φ) = ‖R‖ = ‖m‖(BX#) ≤ ε +‖T‖PIL ∞ , where φ ∈ Lip0(X, L∞(µ)), is the Lipschitz embedding defined by φ = j∞◦ιX. The required inequality follows and the second equality follows in a similar way. � 4. Some relations of Lipschitz Pietsch-p-integral operators with other Lipschitz operator ideals. 4.1. Lipschitz p-summing operators. The definition of the Lipschitz p- summing operators below was first given by Farmer and Johnson in [13]. Definition 4.1. For a pointed metric space X and a Banach space E, the mapping T ∈ Lip0(X, E) is called Lipschitz p-summing, 1 ≤ p < ∞, if there exists a constant C > 0 such that for all x1, . . . , xn, x ′ 1, . . . , x ′ n in X, (4.1) n∑ i=1 ‖T (xi) −T (x ′ i)‖ p ≤ Cp sup f∈B X# n∑ i=1 |f(xi)−f(x ′ i)| p. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 380 Lipschitz integral operators represented by vector measures In this case we put πLp (T ) = inf {C : satisfying (4.1)}. The set of all Lipschitz p-summing operators from X to E is denoted by ΠLp (X, E). It is well known that (ΠLp , π L p (·)) is a Banach Lipschitz operator ideal (see [1, Proposition 2.5]). We can establish the following comparison between the classes of Lipschitz Pietsch-p-integral operators and Lipschitz p-summing operators. Proposition 4.2. Let 1 ≤ p < ∞. Every Lipschitz Pietsch-p-integral operator T : X −→ E is Lipschitz p-summing with πLp (T ) ≤‖T‖PIL p . Proof. If T ∈ PILp (X, E), for ε > 0 we choose a typical Lipschitz Pietsch-p- integral factorization T = A◦ ip ◦B : X B −→ L∞(µ) ip −→ Lp(µ) A −→ E, with ‖A‖Lip(B) ≤ ε + ‖T‖PIL p . The mapping ip is linear p-summing with πp(ip) = 1 (see [10, Page 40]). Then it is Lipschitz p-summing with π L p (ip) = 1 (see [13, Theorem 2]). By the ideal property concerning the Lipschitz operator ideal ΠLp , we have that T ∈ Π L p (X, E) and π L p (T ) ≤‖A‖Lip(B) ≤ ε+‖T‖PIL p . � 4.2. Lipschitz Grothendieck-p-integral operators. The notion of Lips- chitz Grothendieck-p-integral operators (p ≥ 1) from a pointed metric space X into a Banach space E was introduced by Jiménez-Vargas et al. in [15] (under the name of strongly Lipschitz p-integral operators). The mapping T ∈ Lip0(X, E) is Lipschitz Grothendieck-p-integral (in sym- bols T ∈ GILp (X, E)) if JE ◦ T ∈ PI L p (X, E ∗∗), where JE : E −→ E ∗∗ is the canonical injection. The class (GILp ,‖·‖GIL p ) is a Banach Lipschitz operator ideal where ‖T‖GILp = ‖JE ◦T‖PILp (see [3, Remark 4.3 and Proposition 4.8]). It is immediate that PILp (X, E) ⊂ GI L p (X, E) and ‖T‖GIL p ≤ ‖T‖PIL p for all T ∈PILp (X, E). The proof of the next result is an easy adaptation of [5, Proposition 3.3]. Proposition 4.3. If the Banach space E is norm one complemented in E∗∗ (in particular, if E is a dual Banach space), then GILp (X, E) ⊂ PI L p (X, E) and ‖T‖GIL p = ‖T‖PIL p for all T ∈GILp (X, E). 4.3. Strongly Lipschitz p-nuclear operators. Chen and Zheng in [7] intro- duced the concept of strongly Lipschitz p-nuclear operators. For a pointed met- ric space X and a Banach space E, a mapping T ∈ Lip0(X, E) is strongly Lip- schitz p-nuclear (1 ≤ p < ∞) if there exist B ∈ Lip0(X, ℓ∞) and A ∈L(ℓp, E) and a diagonal operator Mλ ∈ L(ℓ∞, ℓp) induced by λ = (λi)i≥1 ∈ ℓp (i.e. Mλ((ξi)i≥1) = (λiξi)i≥1) such that T = A◦Mλ ◦B : X B −→ ℓ∞ Mλ −→ ℓp A −→ E. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 381 E. Dahia and K. Hamidi The Banach space of all these mappings is denoted by SNLp (X, E) and the norm is defined by ‖T‖SN L p = inf ‖A‖‖Mλ‖Lip(B), where the infimum is taken over all the above factorizations. Proposition 4.4. Every strongly Lipschitz p-nuclear operator is Lipschitz Pietsch- p-integral. Moreover, ‖T‖PILp ≤‖T‖SN Lp for all T ∈SNLp (X, E). Proof. Given ε > 0, take T ∈ SNLp (X, E) with the above factorization such that ‖A‖‖Mλ‖Lip(B) ≤ ε +‖T‖SN L p . In this case, ℓ∞ and ℓp are the spaces L∞(µ) and Lp(µ) with µ the counting measure on N respectively and Mλ : L∞(µ) −→ Lp(µ) is the multiplication operator induced by λ ∈ Lp(µ) (i.e. Mλ(f) = λ.f). Use [10, Page 111] to see that Mλ is a Pietsch-p-integral linear operator and ‖Mλ‖PIp = ‖Mλ‖ and then it is Lipschitz Pietsch-p-integral with ‖Mλ‖PILp ≤ ‖Mλ‖ (by Remark 2.2). In view of the ideal property of PILp , we are done. � 4.4. Lipschitz weakly compact operators. The definition of Lipschitz weakly compact operators is due to Jiménez-Vargas et al. ([15]). Definition 4.5. Let X be a pointed metric space and let E be a Banach space. The mapping T ∈ Lip0(X, E) is called Lipschitz weakly compact if the set { T (x)−T (x′) d(x,x′) : x, x′ ∈ X, x 6= x′ } is relatively weakly compact in E. 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