Interpolation of Ideal Measures by Abstract K and J Spaces

Transcription

1 Acta Mathematica Sinica, English Series Aug., 2007, Vol. 23, No. 8, pp Published online: June 21, 2007 DOI: /s Interpolation of Ideal Measures by Abstract K J Spaces Luz M. FERNÁNDEZ-CABRERA Sección Departamental de Matemática Aplicada, Escuela de Estadística, Universidad Complutense de Madrid, Madrid, Spain luz fernez-c@mat.ucm.es Antón MARTÍNEZ Departamento de Matemática Aplicada I, E.T.S. Ingenieros Industriales, Universidad de Vigo, Vigo. Spain antonmar@uvigo.es Abstract We work with the abstract K J interpolation method generated by a sequence lattice Γ. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Γ. Keywords real interpolation, ideal measures, operator ideals, shift operators MR2000 Subject Classification 46B70, 47B10 1 Introduction Interpolation theory is a useful tool in a number of the aspects of operator theory, as the factorization property. Let us recall that an operator ideal I is said to possess the factorization property if each operator T of I factors through a space E whose identity map I E belongs to I. By using the classical real interpolation method A 0,A 1 θ;q, Heinrich [1] showed a sufficient condition for a general closed operator ideal to possess the factorization property. Previous results on this problem are due to Davis, Figiel, Johnson Pelczyński [2] to Beauzamy [3]. More recently, Blanco, Kaijser Ransford [4] have also used interpolation theory to establish that every weakly compact homomorphism between Banach algebras factors through a reflexive Banach algebra, but this time the classical real method is not enough see [5] they need to work with the general real method A 0,A 1 Γ, that is, the interpolation method which is defined by replacing the usual weighted L q norm of A 0,A 1 θ;q by a more general lattice norm Γ. The general real method plays an important role in interpolation theory as can be seen in the monographs papers by Peetre [6], Brudnyǐ Krugljak [7], Cwikel Peetre [8] Nilsson [9 10]. Cobos, Manzano the present authors investigated in [11 12] the behaviour by the general real method of closed operator ideals. As effect they have shown sufficient conditions on I such that, if T L Ā, B T : A 0 A 1 B 0 + B 1 belongs to I, then the interpolated operator T :A 0,A 1 Γ B 0,B 1 Γ also belongs to I. In the present paper we deal with quantitative results. Our aim is to estimate how far the interpolated operator is from the ideal. In the special case of the classical real method, this problem has attracted the attention of a number of authors. Logarithmically convex inequalities Received October 28, 2005, Accepted December 21, 2006 The authors have been supported in part by the Spanish Ministerio de Educación y Ciencia MTM

2 1358 Fernández-Cabrera L. M. Martínez A. like βt A0,A 1 θ;q,b 0,B 1 θ;q cβt A0,B 0 1 θ βt A1,B 1 θ 1.1 have been established by Edmunds Teixeira [13], Cobos, Fernández-Martínez Martínez [14] for the measure of non-compactness; Aksoy Maligra [15], Cobos Martínez [16 17], Kryczka, Prus Szczepanik [18] for the measure of weak non-compactness; Cobos, Manzano Martínez [19] for general ideal measures. Convexity inequalities of the kind 1.1 have no meaning in the general setting. We should look for inequalities of a different shape, involving some features of the lattice Γ. Realizing the real method in a discrete way, that is, working with lattices of sequences Γ with Z as an index set, qualitative results of [11 12] estimates known for the measure of non-compactness see [20 21] suggest that shift operators in Γ should play an important role in this question. We follow this hint we establish interpolation formulae for ideal measures involving a function defined by the norms of shift operators in Γ. The organization of the paper is as follows. In Section 2, we recall some basic concepts on operator theory we introduce ideal measures. In Section 3, we review the notion of abstract K J spaces. We also establish there estimates for ideal measures when one of the Banach couples reduces to a single Banach space. The general case is studied in Section 4. With this aim, we suppose that the operator ideal satisfies the so-called Σ Γ -property. Finally, in Section 5, we investigate the case when the information on the operator refers to the restriction from the intersection into the sum. Specifying the results of Section 5 for the case of the classical real method, we get formulae that improve those established in [17]. 2 Ideal Measures In what follows, E, F denote Banach spaces L E,F sts for the collection of all bounded linear operators from E into F, endowed with the usual operator norm. We put U E for the closed unit ball of E, E for the dual space of E. Wewritel 1 U E for the Banach space of all absolutely summable families of scalars {λ x } x UE with U E as an index set. By l U F we designate the Banach space of all bounded families of scalars indexed by the elements of U F. The operator Q E : l 1 U E E defined by Q E {λ x } = x U E λ x x is a metric surjection, the operator J F : F l U F, given by J F y = { f,y } f UF, is an isometric embedding. An operator ideal I is a method of ascribing to each pair E,F of Banach spaces a linear subspace I E,F ofl E,F such that I E,F contains the finite rank operators, it satisfies that STR I G, V whenever R L G, E, T I E,F S L F, V. The ideal I is called closed if I E,F is a closed subspace of L E,F for all Banach spaces E F.WesaythatI is surjective if, for every T L E,F, it follows from TQ E I l 1 U E,FthatT I E,F. An ideal I is called injective if, whenever T L E,F in addition J F T I E,l U F, then it follows that T I E,F. Compact operators weakly compact operators are examples of surjective injective closed operator ideals. Other examples are Rosenthal operators, Banach Saks operators Asplund operators also referred to as decomposing operators or dual Radon Nikodým operators. The class of strictly singular operators is an ideal which is closed injective but it is not surjective. On the other h, the ideal of strictly cosingular operators is closed surjective but it is not injective. We refer to [22], to [1] [23] for more details about operator ideals. Related to an ideal I we may consider the following functionals: For T L E,F, the outer measure T = T E,F is the infimum of all σ>0 such that T U E σu F + RU Z for some Banach space Z some operator R I Z, F. The inner measure T = T E,F is the infimum of all σ>0such that, for some Banach space Z some operator R I E,Z, the inequality Tx F σ x E + Rx Z

3 Real Interpolation Ideal Measures 1359 holds for all x E. Clearly, if T I then T = T = 0. As for the converse, it is known that If I is surjective closed, then T = 0 if only if T I ; If I is injective closed, then T = 0 if only if T I. Therefore, if the ideal I is closed surjective, we may use the functional to measure the deviation of T from I. IfI is closed injective, the suitable functional to measure the extent to which T fails to belong to I, is. We refer to [19] for relevant references more details on ideal measures. When I = K, the ideal of compact operators, then γ K T coincides with the ball measure of non-compactness, that is to say, γ K T is the infimum of the set of all numbers σ>0 for which there is a finite subset {f 1,...,f k } F such that k T U E {b j + σu F }. j=1 The measure β K T turns out to be the infimum of all μ>0 such that there exists a subspace M of E with finite codimension such that Tx F μ x F for all x M. Another useful measure of non-compactness is T K =inf{ T K : K K E,F}. We have maxγ K T,β K T T K if F has the compact approximation property with constant C, then T K Cγ K T,T LE,F see [24, Thm ]. We refer to [25] for recent results on the K -measure of the noncompactness of some classical operators. Measures γ K β K are equivalent. Namely, we have 1 2 γ K T β K T 2γ K T,T LE,F see [24, Thm ]. However, if we take I as the ideal W of weakly compact operators, then γ W β W are not equivalent. Indeed, there are a Banach space E a sequence of operators {T n } LE,c 0 such that β W T n = 1 but γ W T n =1/n see [26]. Returning to the general case, we always have max { T, T } T. Ideal measures are submultiplicative TR T R, TR T R, for any isometric embedding j : F V any metric surjection π : G E, they satisfy T = jt T = Tπ. 2.1 Moreover, operators J F Q E have the following minimal properties: J F T =min{ jt:j L F, V is an isometric embedding} T, 2.2 TQ E =min{ Tπ:π L G, E is a metric surjection} T ; 2.3 see [19, p ]. Properties 2.1, yield J F T = J F TQ E TQ E = J F TQ E J F T. That is to say, J F T = TQ E. 2.4 The value given by 2.4 is another interesting ideal measure.

4 1360 Fernández-Cabrera L. M. Martínez A. Definition 2.1 Clearly, Let I be an operator ideal. For any T L E,F, weput T = J F T = TQ E. T min { T, T } T. Moreover, for any isometric embedding j L F, V any metric surjection π L G, E, it follows from 2.1 Definition 2.1 that jtπ= T. We shall work with the functional in Sections 4 5. Note that if I is surjective, injective closed, then T = 0 if only if T I. 3 Abstract K J Spaces Let Ā = A 0,A 1 be a Banach couple, i.e., two Banach spaces A j, j = 0, 1, which are continuously embedded in a common Hausdorff topological vector space. Then we can form their sum ΣĀ =A 0 + A 1, intersection ΔĀ =A 0 A 1, these spaces become Banach spaces when endowed with the norms a A0 +A 1 = K1,a a A0 A 1 = J1,a, where for t>0, Kt, a =Kt, a; Ā =inf{ a 0 A0 + t a 1 A1 : a = a 0 + a 1,a j A j }, a A 0 + A 1, Jt, a =Jt, a; Ā =max{ a A 0,t a A1 }, a A 0 A 1. In what follows, Γ denotes a Banach space of real-valued sequences with Z as an index set. We assume that Γ contains all sequences with only finitely many non-zero coordinates that Γisalattice, that is, whenever ξ m μ m for each m Z {μ m } Γ, then {ξ m } Γ {ξ m } Γ {μ m } Γ. We say that Γ is K-non-trivial if {min1, 2 m } Γ. 3.1 The lattice Γ is called J-non-trivial if { } sup min1, 2 m ξ m : {ξ m } Γ 1 <. 3.2 m= More information on conditions can be found in [9] from where we have taken the terminology. Given a Banach couple Ā =A 0,A 1 ak-non-trivial sequence space Γ, the K-space Ā Γ;K =A 0,A 1 Γ;K consists of all a A 0 + A 1 such that {K2 m,a} Γ. The norm of ĀΓ;K is given by a ĀΓ;K = {K2 m,a} Γ. If Γ is J-non-trivial, the J-space ĀΓ;J =A 0,A 1 Γ;J is formed by all those a A 0 + A 1 which can be represented by a = m= u m convergence in A 0 + A 1, where {u m } A 0 A 1 {J2 m,u m } Γ. We put a ĀΓ;J =inf { {J2 m,u m } Γ : a = m= u m }. The spaces ĀΓ;K ĀΓ;J are Banach spaces. Note that, if {J2 m,u m } Γ, then the series m= u m is absolutely convergent in A 0 + A 1. Indeed, using 3.2, we have u m A0 +A 1 min1, 2 m J2 m,u m m= m= { sup m= } min1, 2 m ξ m : {ξ m } Γ 1 {J2 m,u m } Γ <.

5 Real Interpolation Ideal Measures 1361 For 0 <θ<11 q, the classical real interpolation space A 0,A 1 θ,q coincides with the abstract K- J-spaces generated by Γ = l q 2 θm, the space l q with the weight {2 θm }, A 0,A 1 lq 2 θm ;K =A 0,A 1 lq 2 θm ;J =A 0,A 1 θ,q see [7, 27, 28]. If Γ is K-J-non-trivial, it follows from the fundamental lemma of interpolation theory see [27, Lemma 3.3.2], that ĀΓ;K ĀΓ;J. Here means continuous inclusion. In general K- J-spaces do not agree. A necessary sufficient condition for equality can be found in [9, Lemma 2.5]. Let B =B 0,B 1 be another Banach couple. We write T L Ā, B tomeanthatt is a linear operator from A 0 + A 1 into B 0 + B 1 whose restriction to each A j defines a bounded operator from A j into B j for j =0, 1. We set T Ā, B =max{ T A0,B 0, T A1,B 1 }. If one of the couples reduces to a single Banach space, i.e., A 0 = A 1 = A or B 0 = B 1 = B, we put T L A, B T L Ā, B, respectively. Given T L Ā, B, the restrictions T :A 0,A 1 Γ;K B 0,B 1 Γ;K T :A 0,A 1 Γ;J B 0,B 1 Γ;J are bounded with norms less than or equal to T Ā, B. A better estimate can be obtained by using the norms of the shift operators on Γ see [29], [20 21]. Given k Z, theshiftoperator τ k is defined by τ k ξ = {ξ m+k } m Z for ξ = {ξ m } m Z. Subsequently we assume that τ k is bounded in Γ for all k Z, the norms satisfy lim n 2 n τ n Γ,Γ = 0 lim τ n Γ,Γ = n We put M 1 =max1, τ 1 Γ,Γ. Definition 3.1 Let f :0, 0, be the function defined by ft = τ [log t] Γ,Γ. Here, the logarithm is taken with base 2 [ ] is the greatest integer function. Using 3.3, we obtain M 2 =sup{ft :0<t 1} =sup{ τ n Γ,Γ : n 0} <, { } ft M 3 =sup :1 t< =sup { 2 n τ n Γ,Γ : n 0 } <. t Other properties of f are given in the next lemma. They are consequences of 3.3, definitions of M 1, M 2, M 3 the fact that τ m+k Γ,Γ τ m Γ,Γ τ k Γ,Γ, m,k Z. Lemma 3.2 The function f satisfies : ft i lim t 0 ft =0 lim t t =0; ii fst M 1 fsft for all t, s > 0, f2 m s f2 m fs for all s>0 m Z; iii If s<t, then fs M 1 M 2 ft ft fs t M 1 M 3 s. Next we estimate the norms of interpolated operators by using the function f. Lemma 3.3 Let Γ be a sequence space satisfying 3.3. Let Ā =A 0,A 1 B =B 0,B 1 be Banach couples, let T L Ā, B. i If Γ is K-non-trivial, then T ĀΓ;K, B Γ;K =0 if T Aj,B j =0 for j =0or 1, T ĀΓ;K, B Γ;K c T A0,B 0 f T A1,B 1 / T A0,B 0 otherwise.

6 1362 Fernández-Cabrera L. M. Martínez A. ii If Γ is J-non-trivial, then T ĀΓ;J, B Γ;J =0 if T Aj,B j =0 for j =0or 1, T ĀΓ;J, B Γ;J c T A0,B 0 f T A1,B 1 / T A0,B 0 otherwise. Here, c = f2. Proof Let k j > T Aj,B j for j =0, 1taker Z such that 2 r 1 k 1 /k 0 < 2 r.givenany a ĀΓ;J any J-representation a = m= u m,sincej2 m,tu m+r k 0 J2 m+r,u m+r a = m= u m+r, weget Ta B Γ;J {J2 m,tu m+r } Γ k 0 {J2 m+r,u m+r } Γ k 0 τ r Γ,Γ {J2 m,u m } Γ k 0 f2k 1 /k 0 {J2 m,u m } Γ ck 0 fk 1 /k 0 {J2 m,u m } Γ. It follows that T ĀΓ;J, B Γ;J ck 0 fk 1 /k 0. If T Aj,B j = 0 for j = 0 or 1, letting k j 0 using Lemma 3.2/i, we derive that T ĀΓ;J, B Γ;J =0. If T Aj,B j > 0forj =0, 1, taking any ε>0 choosing k j = 1 + ε T Aj,B j,weget T ĀΓ;J, B Γ;J c 1 + ε T A0,B 0 f T A1,B 1 / T A0,B 0. Now letting ε 0 we arrive at the inequality in statement ii. The proof of i is given in [20, Lemma 4.3]. Example 3.4 Let 1 q take Γ = l q 1/g2 m, where g :0, 0, is a function parameter, that is, a positive function such that gt increases from 0 to, gt/t decreases from to0,foreveryt > 0, s g t = sup{gts/gs :s>0} is finite s g t =omax1,t as t 0t. InthiscaseitisknownthatK- J-spaces coincide they are equal to the real interpolation space with function parameter g A 0,A 1 lq 1/g2 m ;K =A 0,A 1 lq 1/g2 m ;J =A 0,A 1 g,q see [6], [30 32]. In particular, for gt =t θ with 0 <θ<1, we recover the classical real interpolation space A 0,A 1 θ,q. Since τ k lq 1/g2 m,l q 1/g2 m s g 2 k, working with spaces A 0,A 1 g,q, we may use s g instead of f. Notethats g satisfies all statements of Lemma 3.2 with M 1 = M 2 = M 3 =1. ReturningtotheabstractK- J-spaces, let A = ĀΓ;K or ĀΓ;J write ψ A t =ψ A t; Ā =sup{kt, a :a A, a A =1}, ρ A t =ρ A t; Ā =inf{jt, a :a A 0 A 1, a A =1}. These functions can be controlled by f as we show next. We put e s = {δm} s m Z, where δm s is the Kronecker delta. Lemma 3.5 Let Γ be a sequence space satisfying 3.3 let Ā =A 0,A 1 be a Banach couple. i If Γ is K-non-trivial, then ψ ĀΓ;K t 2 e 0 Γ ft; ii If Γ is J-non-trivial, then 1/ρ ĀΓ;J t 2 e 0 Γ f1/t for all t>0. Proof For each a U ĀΓ;K each s Z, wehave K2 s,a e s Γ {K2 m,a} Γ 1= e 0 Γ = τ se s Γ τ s Γ,Γ e s Γ. e 0 Γ e 0 Γ e 0 Γ Thus, K2 s,a τ s Γ,Γ / e 0 Γ. Hence, for any t>0, we get Kt, a 2K2 [log t],a 2 ft. e 0 Γ This proves i. To establish ii, take any a A 0 A 1 given any t>0, put s =logt if t =2 m for some m Z s = [log t] + 1 for the remaining values of t. Weobtain a ĀΓ;J J2 s,ae s Γ J2 s,a τ s Γ,Γ e 0 Γ 2Jt, af1/t e 0 Γ.

7 Real Interpolation Ideal Measures 1363 This yields that 1/ρ ĀΓ;J t 2 e 0 Γ f1/t. Lemma 3.5 allows us to apply the results of [33] to the abstract K- J-spaces. In effect we are going to establish estimates for ideal measures in the special case when one of the Banach couples reduces to a single Banach space. The following results are direct consequences of Lemma 3.5 [33, Thms ]: Theorem 3.6 Let Γ be a K-non-trivial sequence space satisfying 3.3 let I be an operator ideal. Suppose Ā = A 0,A 1 is a Banach couple, B is a Banach space T L Ā, B. Then the following hold : a If T Aj,B =0 for j =0or 1, then T ĀΓ;K,B =0; b If T Aj,B > 0 for j =0, 1, then T ĀΓ;K,B 4 e 0 Γ T A0,Bf γi T A1,B. T A0,B Theorem 3.7 Let Γ be a J-non-trivial sequence space satisfying 3.3 let I be an operator ideal. Suppose A is a Banach space, B =B 0,B 1 is a Banach couple T L A, B. Then the following hold : a If T A,Bj =0 for j =0or 1, then T A, B Γ;J =0; b If T A,Bj > 0 for j =0, 1, then βi T A,B1 T A, B Γ;J 2 e 0 Γ T A,B0 f. T A,B0 Sometimes the relevant information on T L Ā, B refers to the restriction T : A 0 A 1 B. Similarly, if T L A, B, we might have better knowledge on the restriction T : A B 0 + B 1 than T : A B j see, for example, [34, 4]. The following lemma is useful in estimating the ideal measures of interpolated operators by means of those other restrictions. Lemma 3.8 Let Ā =A 0,A 1 be a Banach couple put, à =A 0 + A 1,A 0 A 1. i If Γ is K-non-trivial sequence space satisfying 3.3, then ψ ĀΓ;K t; à 4 ft+tft 1 for 0 <t 1; e 0 Γ ii If Γ is J-non-trivial sequence space satisfying 3.3, then 1 ρ ĀΓ;J t; à 4 e 0 Γ t 1 ft+ft 1 for 0 <t 1. Proof According to [35, Thm. 3], for 0 <t 1, max Kt, a; Ā,tKt 1,a; Ā Kt, a; à 2 Kt, a; Ā+tKt 1,a; Ā. 3.4 Therefore, using Lemma 3.5, we get { Kt, a; ψ ĀΓ;K t; à =sup à a ĀΓ;K } : a ĀΓ;K,a 0 2ψ ĀΓ;K t; Ā+tψĀ t 1 ; Ā 4 ft+tft 1. Γ;K e 0 Γ To establish ii we shall use the duality. Let A j be the closure of A 0 A 1 in A j, j =0, 1. Set Ā =A 0,A 1. Since A 0 A 1 = A 0 A 1,wehaveĀΓ;J = Ā Γ;J.SoρĀ t; Ā Γ;J =ρā t; Ā. Γ;J Moreover, ρ ĀΓ;J t; à = ρā t; A Γ;J 0 + A 1,A 0 A 1 because a A = a 0 +A 1 A 0 +A 1 for any a A 0 A 1. Hence, without loss of generality we may do assume that A 0 A 1 is dense in A j for j =0, 1.

8 1364 Fernández-Cabrera L. M. Martínez A. Using the duality between J-K-functionals see [27, Thm ] 3.4, we obtain, for 0 <t 1, { } Jt, a; Ã =sup h, a Kt 1,h;A 0 + A 1, A 0 A 1 : h A 0 A 1 { } h, a =sup t 1 Kt, h; A 0 + A 1,A 0 A 1 : h A 0 A 1 1 { } 2 sup h, a t 1 Kt, h; A 0,A 1 +Kt 1,h; A 0,A 1 : h A 0 A 1 1 [ 1 1 ] 1 t 1 h, a h, a inf 2 h A 0 A 1 Kt, h; A 0,A 1 + inf h A 0 A 1 Kt 1,h; A 0,A 1 = 1 2 [t 1 Jt 1,a; Ā 1 + Jt, a; Ā 1 ] 1. Consequently, by Lemma 3.5, we derive { } 1 a ĀΓ;J =sup ρ ĀΓ;J t; Ã Jt, a; Ã : a A 0 A 1,a 0 { a ĀΓ;J tjt 1,a; Ā + a Ā Γ;J 2sup 2 1 tρ ĀΓ;J t 1 ; Ā + 1 ρ ĀΓ;J t; Ā } Jt, a; Ā : a A 0 A 1,a 0 4 e 0 Γ t 1 ft+ft 1. Suppose now that T L Ā, B leti be an operator ideal. Since T A0 A 1,B T A0 A 1,A 0 +A 1 T A0 +A 1,B T A0 +A 1,B, we have T A0 A 1,B/ T A0 +A 1,B 1. Similarly, for T L A, B, we get T A,B0 +B 1 / T A,B0 B 1 1. Then Lemma 3.8 [33, Thms ] yield the following results: Theorem 3.9 Let Γ be a K-non-trivial sequence space satisfying 3.3 let I be an operator ideal. Suppose Ā = A 0,A 1 is a Banach couple, B is a Banach space T L Ā, B. a If T A0 A 1,B =0,then T ĀΓ;K,B =0; b If T A0 A 1,B > 0,then T ĀΓ;K,B 4 γi T A0 A γ e 0 I T 1,B γi T A0 +A A0 +A 1,Bf + γ Γ T I T 1,B A0 A 1,Bf. A0 +A 1,B T A0 A 1,B Theorem 3.10 Let Γ be a J-non-trivial sequence space satisfying 3.3 let I be an operator ideal. Suppose A is a Banach space, B = B0,B 1 is a Banach couple T L A, B. a If T A,B0 +B 1 =0,then T A, B Γ;J =0; b If T A,B0 +B 1 > 0,then T A, B Γ;J βi T A,B0 +B 4 e 0 Γ T A,B0 B 1 f 1 βi T A,B0 B + β T A,B0 B 1 I T A,B0 +B 1 f 1. T A,B0 +B 1 Writing down Theorem 3.9 for Γ = l q 2 θm, 0 <θ<1, we get an improvement of a result of Cobos Martínez [16, Thm. 3.2], since in their estimate we can replace T Ā,B by the smaller factor T A0 +A 1,B. Similarly, specializing Theorem 3.10 for Γ = l q 2 θm, we obtain an improvement of [16, Thm. 3.3], where we can now replace T A, B by the smaller term T A,B0 B 1.

9 Real Interpolation Ideal Measures General Interpolation Results In this section we estimate ideal measures of interpolated operators in the general case. We recall that a sequence lattice Γ is said to be regular if, for any sequence {φ n } n N Γ with φ n 0, it follows that φ n Γ 0asn. The associated space Γ of Γ consists of all sequences {δ m } for which { } {δ m } Γ =sup δ m ξ m : {ξ m } Γ 1 <. m= Given any sequence space Γ any sequence of Banach spaces {E m },wedenotebyγe m the vector-valued Γ-space defined by ΓE m ={x = {x m } : x m E m x ΓEm = { x m Em } Γ < }. Let Q k :ΓE m E k be the projection Q k {x m } = x k,letp r : E r ΓE m bethe embedding P r x = {δmx}, r where again δm r is the Kronecker delta. The following notion was introduced in [1] [11]: Definition 4.1 An operator ideal I is said to satisfy the Σ Γ -condition if, for any sequences of Banach spaces {E m }, {F m } for any operator T L ΓE m, ΓF m, it follows from Q k TP r I E r,f k for any r, k Z that T I ΓE m, ΓF m. Clearly, if I satisfies the Σ Γ -condition then the identity operator I Γ on Γ must belong to I. Moreover, the ideal I must be closed see [11, Lemma 3.2]. Let I be the ideal of weakly compact operators, Rosenthal operators or Banach Saks operators. Then it is shown in [11, Thms ] that I satisfies the Σ Γ -condition, for any sequence lattice such that I Γ belongs to I. Next we investigate the case of Asplund operators. Recall that T L E,FissaidtobeanAsplund operator if T is a Radon Nikodým operator, that is, for any probability measure μ, T maps each μ-continuous F -valued measure of finite variation into a μ-differentiable E -valued measure see [22] [1] where these operators are referred to as decomposing operators. The following characterization of Asplund operators is useful see [36, Thm ]: Theorem 4.2 Let T L E,F. Then T is Asplund if only if the seminormed space F,ν T D is separable whenever D U E is countable. Here ν T D g =sup{ gtx : x D}, g F. Heinrich [1] showed that Asplund operators satisfy the Σ lq -condition for 1 <q<. The next result refers to the general case. Theorem 4.3 The ideal A of Asplund operators satisfies the Σ Γ -condition for any sequence space Γ such that I Γ A Proof Let {E m }, {F m } be arbitrary sequences of Banach spaces let T L ΓE m, ΓF m be such that Q k TP r A E r,f k for any r, k Z. Fix any r Z take any D U Er countable. Write O k = Q k TP r D. Let Ω k be a countable set that is dense in Fk,τ O k. We claim that the countable set { } Ω= P k h k : h k Ω k,k Z,N N k N is dense in ΓF m,ν TP r D. Indeed, since I Γ A, Γ does not contain a copy of either of l or l 1. Hence, by [37, Thms ], the sequence spaces Γ Γ are regular. It follows that ΓF m =Γ Fm see [38, Prop. 3.1]. Take any g Γ Fmletg m = Q m g Fm. By the regularity of Γ, given any ε>0, there is N N such that g Γ P k g k ε/2 TP r Fm Er,ΓF m. k N

10 1366 Fernández-Cabrera L. M. Martínez A. For each k N, chooseh k Ω k such that ν Ok g k h k ε/4n +2. Write h = k N P kh k. Then h Ω we have ν TP g g h ν P rd TPrD k g k + ν P TP rd kg k h k k N k N k N TP r Er,ΓF m g Γ P k g k + ν Fm Ok g k h k ε. k N This proves our claim shows that TP r A E r, ΓF m for every r Z. Now take any W U ΓEm countable. For each r Z, thesetw r = Q r W U Er is countable. Hence, ΓF m,ν TP r Wr is separable. Let Λr be a countable set that is dense in ΓFm,ν TPrWr. To complete the proof it suffices to show that the countable set { } Λ= P r Q r h r : h r Λ r,r Z,N N r N is dense in ΓF m,ν T W. Take any g ΓF m =Γ Fmanyε>0. Again using the regularity of Γ,wecan find N N such that gt P r Q r gt ε/2. Γ G m r N For each r N, thereish r Λ r such that ν gt h P r Qr W rt ε/4n + 2. Put h = r N P rq r h r.thenh Λ ν W gt ht gt P r Q r gt + ν Γ W P r Q r gt h r T G m r N r N ε/2+ ν gt h P r Qr W rt ε. r N This completes the proof. The next theorem shows the behaviour of ideal measures under interpolation by the K- method. Given two Banach spaces Z 0, Z 1,wedenotebyZ 0 Z 1 l respectively, Z 0 Z 1 l1 the direct sum of Z 0 Z 1,normedby x, y =max x Z0, x Z1 respectively, x, y 1 = x Z0 + x Z1. Theorem 4.4 Let Γ be a K-non-trivial sequence space satisfying 3.3 let I be an operator ideal which satisfies the Σ Γ -condition. Assume that Ā =A 0,A 1, B =B0,B 1 are Banach couples let T L Ā, B. Then the following holds : a If T Aj,B j =0 for j =0or 1, then T ĀΓ;K, B Γ;K =0; b If T Aj,B j > 0 for j =0, 1, then γi T A1,B T ĀΓ;K, B Γ;K c T A0,B 0 f 1, T A0,B 0 where c =2f2. Proof For m Z, letf m be the space B 0 + B 1 endowed with the norm K2 m,. Since the map j : B Γ;K ΓF m defined by jb =...,b,b,b,... is an isometric embedding, it follows from 2.2 the definition of that T ĀΓ;K, B Γ;K = J B Γ;K T jt. 4.1 Take any k j > T Aj,B j, j =0, 1, find r Z such that 2 r 1 k 1 /k 0 < 2 r. By the definition of T Aj,B j, there are a Banach space Z j anoperators j I Z j,b j such that T U Aj kj U Bj + S j UZj,j=0,

11 Real Interpolation Ideal Measures 1367 For m Z, put W m =Z 0 Z 1 l. Given any positive number ε, define S :ΓW m ΓF m by S{x 0,m,x 1,m } = { 1 + εf2 r S 0 x 0,m +2 m k 0 k 1 1 S 1x 1,m }. The operator S is bounded, because S{x 0,m,x 1,m } ΓFm 1 + εf2 r max 1, k 0 max{ S 0 Z0,B 0, S 1 Z1,B 1 } {x 0,m,x 1,m } ΓWm. k 1 Moreover, for each r, s Z, wehave { 0, if r s, Q s SP r x 0,x 1 = 1 + εf2 r S 0 x 0 +2 s k 0 k1 1 S 1x 1, if r = s. So, using the projection operators from W r onto Z 0 Z 1, the definition of the operator ideal, we get that Q s SP r I W r,f r. Therefore, by the Σ Γ -property of I, it follows that S I ΓW m, ΓF m. We claim that jtu ĀΓ;K 1 + εck 0 fk 1 /k 0 U ΓFm + S U ΓWm. 4.3 Indeed, given any 0 a U ĀΓ;K, put d m = d m a =1+εK2 m,a. Since K2 m k 1 /k 0,a K2 m+r,a <d m+r, we can find the decomposition a = a 0,m + a 1,m with a j,m A j, d 1 m+ra 0,m A0 1 2 m k 1 /k 0 d 1 m+r a 1,m A1 1. By 4.2, there exist x j,m U Zj, j =0, 1, so that Ta j,m S j 2 m k 1 /k 0 j d m+r x j,m Bj 2 jm k 0 d m+r, j =0, Let x = { 1 + ε 1 f2 r 1 d m+r x 0,m, 1 + ε 1 f2 r 1 d m+r x 1,m }. Then x ΓWm 1 + ε 1 f2 r 1 {d m+r } Γ 1 + ε 1 f2 r 1 τ r Γ,Γ {d m } Γ 1. Furthermore, by 4.4, K2 m,p m jta Sx Ta 0,m d m+r S 0 x 0,m B0 +2 m Ta 1,m 2 m k 0 k1 1 d m+rs 1 x 1,m B1 2k 0 d m+r. Consequently, jta Sx ΓFm = {K2 m,p m jta Sx} Γ 2k 0 {d m+r } Γ 2k 0 τ r Γ,Γ 1 + ε 2f21 + εk 0 fk 1 /k 0. This establishes 4.3. From , we obtain T ĀΓ;K, B Γ;K 1 + εck 0 fk 1 /k 0, the result wanted follows by proceeding as at the end of Lemma 3.3. Remark 4.5 The functional is essential in Theorem 4.4. Example given in [19], Remark 3.4 shows that the results fail if we replace by. The next theorem refers to interpolation by the J-method. Theorem 4.6 Let Γ be a J-non-trivial sequence space satisfying 3.3 let I be an operator ideal which satisfies the Σ Γ -condition. Assume that Ā =A 0,A 1, B =B0,B 1 are Banach couples let T L Ā, B. Then the following hold : a If T Aj,B j =0 for j =0or 1, then T ĀΓ;J, B Γ;J =0; b If T Aj,B j > 0 for j =0, 1, then βi T A1,B T ĀΓ;J, B Γ;J c T A0,B 0 f 1, T A0,B 0

12 1368 Fernández-Cabrera L. M. Martínez A. where c = f2. Proof Let G m be the space A 0 A 1 normed by J2 m,, m Z. Themap π :ΓG m ĀΓ;J defined by π{u m } = m= u m is a metric surjection. Using 2.3, we have T ĀΓ;J, B Γ;J = TQ ĀΓ;J Tπ. 4.5 Take any k j > T Aj,B j, j =0, 1, choose r Z such that 2 r 1 k 1 /k 0 < 2 r.bythe definition of T Aj,B j, there exist a Banach space Z j an operator S j I A j,z j such that Ta Bj k j a Aj + S j a Zj,a A j,j=0, For m Z,denoteby W m the Banach space Z 0 Z 1 l1,lets :ΓG m ΓW m be the operator assigning to every {u m } ΓG m the sequence S{u m } = { τ r Γ,Γ S 0 u m, 2 m r S 1 u m }. We have S{u m } ΓWm τ r Γ,Γ { S 0 u m Z0 +2 m r S 1 u m Z1 } Γ τ r Γ,Γ max S 0 A0,Z 0, S 1 A1,Z 1 max1, 2 r {u m } ΓGm. Hence, S L ΓG m, ΓW m. Since for each r, s Z, the composition Q s SP r belongs to I G r,w s, the Σ Γ -property of I implies that S I ΓG m, ΓW m. We claim that Tπ{u m } B Γ;J ck 0 fk 1 /k 0 {u m } ΓGm + S{u m } ΓWm, {u m } ΓG m. 4.7 Indeed, given any {u m } ΓG m, we have It follows from 4.6 that u m A0 J2 m,u m, u m A1 2 m r k 1 1 J2 m,u m. k 0 Tu m B0 k 0 J2 m,u m + S 0 u m Z0, Tu m B1 2 r m k 0 J2 m,u m + S 1 u m Z1. Consequently, Tπ{u m } B Γ;J {J2 m,tu m+r } Γ τ r Γ,Γ {J2 m r,u m } Γ τ r Γ,Γ {max Tu m B0, 2 m r Tu m B1 } Γ τ r Γ,Γ {k 0 J2 m,u m + S 0 u m Z0 +2 m r S 1 u m Z1 } Γ τ r Γ,Γ k 0 {u m } ΓGm + S{u m } ΓWm f2k 0 fk 1 /k 0 {u m } ΓGm + S{u m } ΓWm. This proves 4.7. Finally, yield the result. 5 Extreme Estimates In this final section we establish Theorems in the general case. In the special situation of the classical real method, that is, for Γ = l q 2 θm, this question has been investigated by Cobos one of the present authors in [17]. Our techniques use some ideas introduced in [17] but, as we shall show later, specifying our results for the case of the classical real method, we will get improvements of the main results of [17]. Let Ā =A 0,A 1, B =B0,B 1 be Banach couples, for each m Z, put G m = G m Ā =A 0 A 1,J2 m,, F m = F m B =B 0 + B 1,K2 m,. Given any T L Ā, B any operator ideal I,wewrite T J,K =sup { T Gm Ā,F m B : m Z }

13 Real Interpolation Ideal Measures 1369 Clearly, T K,K =sup { T Fm Ā,F m B : m Z }. T J,K T K,K T Ā, B. Moreover, since any two norms of the family {Kt, } t>0 respectively, {Jt, } t>0 areequivalent, we have T J,K = 0 if only if T A0 A 1,B 0 +B 1 =0. The proof of the next approximation result is a minor modification of the proof of [17], Lemma 2.1. Lemma 5.1 Let Ā, B, T I be as above. Assume that Γ is a sequence space. Given any k> T J,K σ> T K,K, there exist Banach spaces {Z m } m Z, {X m } m Z operators R m I Z m,f m B, S m I X m,f m B such that T U Gm Ā ku Fm B + R m U Zm, sup { R m Zm,F m B : m Z } < T U Fm Ā σu Fm B + S m U Xm, sup { S m Xm,F m B : m Z } <. In addition, if I satisfies the Σ Γ -condition then the operators R :ΓZ m ΓF m B S :ΓX m ΓF m B defined by R{z m } = {R m z m }, S{x m } = {S m x m } belong to I. In the next result we estimate T ĀΓ;K, B Γ;K in terms of the restrictions of T from G m Ā into F m B fromf m Ā intof m B. Theorem 5.2 Let Ā, B, Gm, F m T be as before. Let Γ be a K-non-trivial sequence space satisfying 3.3 assume that I is an operator ideal satisfying the Σ Γ -condition. Then we have : a If T A0 A 1,B 0 +B 1 =0,then T ĀΓ;K, B Γ;K =0; b If T A0 A 1,B 0 +B 1 > 0,then γi T J,K γi T K,K T ĀΓ;K, B Γ;K M T K,K f + γ T K,K I T J,K f, T J,K where M =2max{1,f2}. Proof Let ε>0takeanyk> T J,K σ> T K,K, so that k σ. Find r Z, r 0, satisfying 2 r 1 <k/σ 2 r. According to Lemma 5.1, there are sequences of Banach spaces {Z m }, {X m } operators R m I Z m,f m B, S m I X m,f m B such that T U Gm Ā ku Fm B + R m U Zm, m Z, 5.1 T U Fm Ā σu Fm B + S m U Xm, m Z. Moreover, operators R :ΓZ m ΓF m B, S :ΓX m ΓF m B defined by R{z m } = {1 + ε2 r+1 τ r Γ,Γ + τ r+1 Γ,Γ R m z m }, S{x m } = {1 + ε τ r Γ,Γ +2 r τ r+1 Γ,Γ S m x m }, belong to I. Put Y =ΓZ m ΓX m l let L : Y ΓF m B be the operator assigning to every pair {z m } m Z, {x m } m Z the sequence L{z m }, {x m }=R{z m } + S{x m }. Using the projection operators from Y onto ΓZ m ΓX m, the definition of the operator ideal, we obtain that L I Y,ΓF m B. We are going to show that jtu ĀΓ;K 1 + εmkfσ/k+σfk/σu ΓFm B + LU Y. 5.2 Given any 0 a U ĀΓ;K put d m = d m a =1+εK2 m,a. Since K2 m k/σ, a K2 m r,a <d m r K2 m σ/k, a K2 m+r+1,a <d m+r+1, we can find decompositions a = a 0,m + a 1,m = a 0,m + a 1,m with a j,m,a j,m A j

14 1370 Fernández-Cabrera L. M. Martínez A. a j,m Aj 2 m σ j d m r, a k j,m Aj 2 m k j d m+r+1,j=0, 1. σ Let b m = a 0,m a 0,m = a 1,m a 1,m, m Z. Thenb m A 0 A 1 with { J2 m,b m max d m+r+1 + d m r, σ k d m r + k } σ d m+r+1 d m+r+1 + σ k d m r. Put ψ m = d m+r+1 +σ/kd m r let c m = a b m.wehave K2 m,c m a 0,m A0 +2 m a 1,m A1 d m r + k σ d m+r+1. Write φ m = d m r +k/σd m+r+1.thena = b m + c m with J2 m,b m ψ m, K2 m,c m φ m. 5.3 By 5.1, for each m Z, therearez m U Zm, x m U Xm, such that K2 m,t m b m R m ψ m z m kψ m, K2 m,t m c m S m φ m x m σφ m. 5.4 Let z ={1 + ε 1 τ r+1 Γ,Γ +2 r+1 τ r Γ,Γ 1 ψ m z m }, x ={1 + ε 1 τ r Γ,Γ +2 r τ r+1 Γ,Γ 1 φ m x m }. Then z ΓZm 1 + ε 1 τ r+1 Γ,Γ +2 r+1 1 τ r Γ,Γ {d m+r+1 + σ m r} k d 1 + ε 1 τ r+1 Γ,Γ +2 r+1 1 τ r Γ,Γ τ r+1 Γ,Γ 1 + ε+ σ k τ r Γ,Γ 1 + ε 1. Similarly x ΓXm 1, so z, x U Y. Finally, using 5.4, we obtain jta Lz, x ΓFm B {K2 m,tb m R m ψ m z m } Γ + {K2 m,tc m S m φ m x m } Γ {kψ m } Γ + {σφ m } Γ =2 {kψ m } Γ 2k τ r+1 Γ,Γ 1 + ε+σ τ r Γ,Γ 1 + ε 21 + εmax{1,f2}kfσ/k+σfk/σ. This establishes 5.2. Since T ĀΓ;K, B Γ;K jt, letting ε 0, we conclude that T ĀΓ;K, B Γ;K Mkfσ/k+σfk/σ. Now the result follows. Specializing the theorem to the case of the real method with a function parameter Example 3.4 we get the following result: Theorem 5.3 Let Ā, B, G m, F m T be as before. Assume that g is a function parameter, 1 q let I be an operator ideal satisfying the Σ lq -condition. Then the following hold : a If T A0 A 1,B 0 +B 1 =0,then T Āg,q, B g,q =0; b If T A0 A 1,B 0 +B 1 > 0,then γi T J,K γi T K,K T Āg,q, B g,q 2s g 2 T K,K s g + γ T K,K I T J,K s g. T J,K In the special case of the classical real method, that is, for gt =t θ with 0 <θ<1, we have s g t =t θ Theorem 5.3/b yields T Āθ,q, B θ,q 8 T J,K Θ T K,K 1 Θ. 5.5 Here Θ = minθ, 1 θ. Γ

15 Real Interpolation Ideal Measures 1371 Inequality 5.5 improves [17, Thm. 2.2], because it replaces in [17] the term T Ā, B by the smaller term T K,K. We notice that a finite number of restrictions is not enough to dominate T ĀΓ;K, B Γ;K ; see [17, Example 3.4]. In order to establish the corresponding result for J-spaces, we put T J,K =sup { T Gm Ā,F m B : m Z } T J,J =sup { T Gm Ā,G m B : m Z }. We have T J,K T J,J T Ā, B. Moreover, T J,K = 0 if only if T A0 A 1,B 0 +B 1 =0. We shall use the duality in the arguments, so we assume that the sequence lattice Γ is regular see Section 4. We designate by Γ the associated space of Γ. By [38, Prop. 3.1], the regularity of Γ yields that Γ =Γ. The space Γ = {{ξ m } m Z : {ξ m } m Z Γ } normed by {ξ m } Γ = {ξ m } Γ is a lattice contains all sequences with only finitely many non-zero coordinates. It is not hard to check that if Γ is J-non-trivial then Γ is K-non-trivial. Shift operators satisfy τ r Γ,Γ = τr Γ,Γ = τ r Γ,Γ = τ r Γ,Γ. 5.6 Given a Banach couple Ā =A 0,A 1 wewrite,asinsection3,a j for the closure of A 0 A 1 in A j, j =0, 1, we set Ā =A 0,A 1. The couple Ā is said to be regular if A j = A j for j =0, 1. If Ā is regular, the dual pair Ā =A 0,A 1 is also a Banach couple. The following formula holds for regular couples see [8, Thm. 3.2 p. 49]: A 0,A 1 Γ;J =A 0,A 1 Γ ;K. 5.7 Given any Banach couple Ā =A 0,A 1, it is clear that G m Ā = A 0 A 1,J2 m, ; Ā = G m Ā, m Z, so ĀΓ;J = Ā Γ;J. Furthermore, for any a A 0 A 1, K2 m,a; Ā =K2 m,a; Ā, m Z. It follows that T Gm Ā,F m B = T Gm Ā,F m B, 5.8 T Gm Ā,G m B = T Gm Ā,G m B, for any T L Ā, B any operator ideal I. We now prove the result for J-spaces. Theorem 5.4 Let Ā, B, G m, F m T be as before. Let Γ be a J-non-trivial sequence space satisfying 3.3 assume that I is an operator ideal satisfying the Σ Γ -condition. Then the following hold : a If T A0 A 1,B 0 +B 1 =0,then T ĀΓ;J, B Γ;J =0; b If T A0 A 1,B 0 +B 1 > 0,then βi T J,K βi T J,J T ĀΓ;J, B Γ;J M T J,J f + β T J,J I T J,K f, T J,K where M =2max{1,f2}. Proof According to 5.8, without loss of generality we may do assume that Ā B are regular couples. Let ε>0takeanyk> T J,K σ> T J,J, with k σ. There are Banach spaces {Z m }, {X m } operators R m I G m Ā,Z m, S m I G m Ā,X m such that Ta Fm B k a Gm Ā + R m a Zm, a G m Ā, 5.9

16 1372 Fernández-Cabrera L. M. Martínez A. Ta Gm B σ a Gm Ā + S m a Xm, a G m Ā. By [39, Lemma 1.1], this is equivalent to T U Fm B ku G m Ā + R mu Z m, T U Gm B σu G m Ā + S mu X m. We can procced as in the proof of [17, Lemma 2.1], find Banach spaces {W m }, {Y m } bounded linear operators i m L W m,zm, j m L Y m,xm such that sup { Rmi m Wm,G m Ā : m Z} <, 5.10 sup { Smj m Ym,G m Ā : m Z} <, T U Fm B ku G m Ā + R mi m U Wm, T U Gm B σu G m Ā + S mj m U Ym. By [27, Thm ] [38, Prop. 3.1], we know that G m Ā = F m Ā,F m B = G m B ΓG m Ā =Γ F m Ā. Moreover, π :ΓGm Ā ĀΓ;J = j : Ā Γ ;K Γ F m Ā. Repeating the arguments in the proof of Theorem 5.2, we derive π T U B Γ;J 1 + εmσfk/σ+kfσ/ku ΓG m Ā + LU W Here W =Γ W m Γ Y m l L L W, ΓG m Ā is the operator defined by L{w m }, {y m }=R{w m } + S{y m },where R{w m } = {1 + ε2 r+1 τ r Γ,Γ + τ r+1 Γ,Γ Rmi m w m }, S{y m } = {1 + ε τ r Γ,Γ +2 r τ r+1 Γ,Γ Smj m y m }, r Z satisfies 2 r 1 <k/σ 2 r. Write λ =1+ε 2 r+1 τ r Γ,Γ + τ r+1 Γ,Γ, μ =1+ε τ r Γ,Γ +2 r τ r+1 Γ,Γ. If z m Z m, let ẑ m be the image of z m in Zm by the canonical embedding. Let V : ΓG m Ā ΓW m be the operator defined by V {u m } = {λi m R m u m }. If follows from 5.10 that V L ΓG m Ā, ΓW m. Moreover, since I satisfies the Σ Γ -condition R m I G m Ā,Z mforeachm Z, we obtain that V I ΓG m Ā, ΓW m. Similarly, for the operator O :ΓG m Ā ΓY m defined by O{u m } = {μjmŝ m u m }, wederivethat O I ΓG m Ā, ΓY m. Let D :ΓG m Ā ΓW m ΓYm l1 be the operator assigning to every sequence {u m } ΓG m Ā the couple D{u m} =V {u m },O{u m }. Clearly, D I ΓG m Ā, ΓWm ΓY m l 1. Put η =1+εMσfk/σ+kfσ/k. We claim that Tπ{u m } B Γ;J η {u m } ΓGm Ā + D{u m } ΓW m ΓYm l 1, {u m } ΓG m Ā Indeed, given any {u m } ΓG m Ā, by the Hahn Banach theorem there exists h U B Γ;J such that Tπ{u m } B Γ;J = h, T π{u m } = π T h, {u m }. According to 5.11, we can find l U ΓGm Ā, w = {w m} U Γ W m y = {y m } U Γ Y m such that π T h = ηl + Rw + Sy = ηl + λ{r mi m w m } + μ{s mj m y m }.

17 Real Interpolation Ideal Measures 1373 Consequently, Tπ{u m } B Γ;J η l, {u m } + λ {R mi m w m }, {u m } + μ {S mj m y m }, {u m } η {u m } ΓGm Ā + λ {i m R m u m }, {w m } + μ {jmŝ m u m }, {y m } η {u m } ΓGm Ā + D{u m } ΓW m ΓYm l 1. Since T ĀΓ;J, B Γ;J Tπ, if follows from 5.12 that T ĀΓ;J, B Γ;J 1 + εmσfk/σ+kfσ/k. This yields the result. Writing down Theorem 5.4 for the case of the real method with a function parameter Example 3.4, we derive the following result: Theorem 5.5 Let Ā, B, G m, F m T be as before. Assume that g is a function parameter, 1 q let I be an operator ideal satisfying the Σ lq -condition. Then the following hold : a If T A0 A 1,B 0 +B 1 =0,then T Āg,q, B g,q =0; b If T A0 A 1,B 0 +B 1 > 0,then βi T J,K βi T J,J T Āg,q, B g,q 2s g 2 T J,J s g + β T J,J I T J,K s g. T J,K For gt =t θ with 0 <θ<1, the space Āg,q coincides with the classical real interpolation space Āθ,q Theorem 5.5/b gives T Āθ,q, B θ,q 8 T J,K Θ T J,J 1 Θ, 5.13 where Θ = minθ, 1 θ. Estimate 5.13 was proved in [17, Thm. 2.3], with T J,J replaced by T Ā, B. Since T J,J T Ā, B, 5.13 improves that result. References [1] Heinrich, S.: Closed operator ideals interpolation. J. Funct. Analysis, 35, [2] Davis, W. J., Figiel, T., Johnson, W. B., Pelczyński, A.: Factoring weakly compact operators. J. Funct. Analysis, 17, [3] Beauzamy, B., d interpolation réels, E.: Espaces d interpolation réels: topologie et géométrie, Lecture Notes in Math., 666, Springer, Berlin, 1978 [4] Blanco, A., Kaijser, S., Ransford, T. J.: Real interpolation of Banach algebras factorization of weakly compact homomorphisms. J. Funct. Analysis, 217, [5] Cobos, F., Fernández-Cabrera, L. M., Martínez, A.: On interpolation of Banach algebras factorization of weakly compact homomorphisms. Bull. Sci. Math., 130, [6] Peetre, J., A theory of interpolation of normed spaces, Lecture Notes, Brasilia, Notes Mat., 39, [7] Brudnyǐ, Y., Krugljak, N.: Interpolation functors interpolation spaces, Vol. 1, North-Holl, Amsterdam, 1991 [8] Cwikel, M., Peetre, J.: Abstract K J spaces. J. Math. Pures Appl., 60, [9] Nilsson, P.: Reiteration theorems for real interpolation approximation spaces. Ann. Mat. Pura Appl., 132, [10] Nilsson, P.: Interpolation of Calderón Ovchinnikov pairs. Ann. Mat. Pura Appl., 134, [11] Cobos, F., Fernández-Cabrera, L. M., Manzano, A., Martínez, A.: Real interpolation closed operator ideals. J. Math. Pures Appl., 83, [12] Cobos, F., Fernández-Cabrera, L. M., Manzano, A., Martínez, A.: On interpolation of Asplund operators. Math. Z., 250, [13] Teixeira, M. F., Edmunds, D. E.: Interpolation theory measures of non-compactness. Math. Nachr., 104, [14] Cobos, F., Fernández-Martínez, P., Martínez, A.: Interpolation of the measure of non-compactness by the real method. Studia Math., 135, [15] Aksoy, A. G., Maligra, L.: Real interpolation measure of weak noncompactness. Math. Nachr., 175, [16] Cobos, F., Martínez, A.: Remarks on interpolation properties of the measure of weak non-compactness ideal variations. Math. Nachr., 208,

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