Duality of measures of non-A-compactness

STUDIA MATHEMATICA 229 (2) (2015) Duality of measures of non-A-compactness by Juan Manuel Delgado (Seville) and C´ andido Pi˜ neiro (Huelva) Abstract. Let Abe a Banach operator ideal. Based on the notion of A-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-A-compactness of an operator. We consider a map χA(respectively, nA) acting on the operators of the surjective (respectively, injective) hull of Asuch that χA(T) = 0 (respectively, nA(T) = 0) if and only if the operator Tis A-compact (respectively, injectively A-compact). Under certain conditions on the ideal A, we prove an equivalence inequality involving χA(T∗) and nAd(T). This inequality provides an extension of a previous result stating that an operator is quasi p-nuclear if and only if its adjoint is p-compact in the sense of Sinha and Karn. 1. Introduction. It is well known that if a bounded subset Aof a Banach space Xis not relatively compact, then there exists ε > 0 such that Acannot be covered by finitely many balls with radii smaller than (or equal to) ε. In this setting, the Hausdorff measure of noncompactness (or the ball measure of noncompactness), χ, is defined for every bounded set A as follows: χ(A) = infnε > 0: A⊂ n [ i=1 xi+εBXo, where BXdenotes the closed unit ball of Xand the infimum is taken over all possible sets of finitely many vectors x1, . . . , xn∈X[11]. Of course, χ(A) vanishes if and only if Ais relatively compact. If Tis a (bounded) linear operator from the Banach space Xto the Banach space Y, the measure of noncompactness of Tcan be defined in a natural way by setting χ(T) = χ(T(BX)). Then χis a seminorm on L(X, Y ), the space of all bounded linear operators from Xto Y, and χvanishes exactly on K(X, Y ), the subspace of L(X, Y ) consisting of all compact opera2010 Mathematics Subject Classification: Primary 47L20, 47B10; Secondary 47H08. Key words and phrases: measure of noncompactness, compact set, operator ideal, p-summing operator, p-compact operator, essential norm. Received 11 June 2014; revised 23 November 2015. Published online 4 January 2016. DOI: 10.4064/sm7984-1-2016 [95] c Instytut Matematyczny PAN, 2015 96 J. M. Delgado and C. Pi˜neiro tors. According to Schauder’s classical theorem, an operator T∈ L(X, Y ) is compact if and only if its adjoint operator T∗is. In 1965, Gol’denˇste˘ın and Markus [12] proved the inequalities 1 2χ(T)≤χ(T∗)≤2χ(T), which, in some sense, may be considered as an extension of Schauder’s theorem. Another extension is obtained if, for instance, the Kuratowski measure of noncompactness, γ, is considered [15]. The definition of γis similar to that of χwith “balls with radii” replaced by “bounded subsets with diameter”. In this case, Astala [2] showed (1.1) γ(T) = γ(T∗) for every T∈ L(X, Y ). Based on Grothendieck’s characterization of relatively compact sets as those sitting inside the convex hull of the norm null sequences, Sinha and Karn [21] introduced a strengthened form of compactness in Banach spaces. Let 1 ≤p < ∞and let p0be the conjugate index of p(i.e., 1/p + 1/p0= 1). A set K⊂Xis said to be relatively p-compact if there exists a p-summable sequence (xn) in Xsuch that A⊂ {Pnαnxn: (αn)∈B`p0}((αn)∈Bc0 if p= 1). The notion of p-compact operator is defined in the obvious way: an operator T∈ L(X, Y ) is said to be p-compact if T(BX) is relatively p-compact in Y. Serrano and the present authors have recently proved the following: T(respectively, T∗) is p-compact if and only if T∗(respectively, T) is quasi p-nuclear [9, Corollary 3.4 and Proposition 3.8]. The main purpose of this paper is to obtain an extension of that result using a sort of measures of noncompactness. Indeed, we consider a positive map χΠd p(respectively, nΠp) acting on Πd p, the ideal of operators with p-summing adjoints (respectively, Πp, the ideal of p-summing operators) vanishing precisely on the class of p-compact operators (respectively, quasi p-nuclear operators). With these maps in hand, an equality like (1.1) relating χΠd pand nΠpis obtained (Corollary 3.13), which provides the desired generalization. Our study is carried out in a more general setting. Given an operator ideal A, the notions of surjective (respectively, injective) A-compactness introduced in [4] (respectively, [23]) are basic to this paper. Section 2 is devoted to the study of the map χA, defined on a certain class of bounded subsets of a Banach space (the so called A-bounded sets), which gives information about the degree of non-A-compactness of these sets in such a way that χAvanishes precisely on the class of (surjectively) A-compact sets. In Section 3, the notion of measure of non-A-compactness is extended to the operator setting using two different (but related) approaches. Indeed, the map χA(respec- Duality of measures of non-A-compactness 97 tively, nA) gives information about the degree of non-A-compactness of an operator, and it vanishes precisely on the class of surjectively (respectively, injectively) A-compact operators. Under certain conditions on the ideal A, we obtain several inequalities involving χAand nAacting on an operator and its adjoint. We show that this approach is different from that appearing in [2] and [24], where the notion of (outer and inner) A-variation of an operator is defined and studied. Finally, we introduce the notion of A-essential norm ρAof an operator in Section 4 and we study the equivalence between χAand ρAunder certain conditions on Xor Y. Our notation is standard. X,Yand Zare always reserved for Banach spaces. A Banach space Xwill be regarded as a subspace of its bidual X∗∗ under the canonical embedding iX:X→X∗∗. We denote the closed unit ball of Xby BX. The Banach space of all bounded linear operators from Xto Yis denoted by L(X, Y ). If Ais an operator ideal, then Addenotes its dual operator ideal, i.e., the one with components Ad(X, Y ) = {T∈ L(X, Y ): T∗∈ A(Y∗, X∗)}. Recall that an operator ideal Ais surjective if, given S∈ A(Z, Y ) and T∈ L(X, Y ), the condition T(BX)⊂S(BZ) implies that T∈ A(X, Y ). For an arbitrary ideal A, the surjective hull Asur of Ais the operator ideal whose components are Asur(X, Y ) = {T∈ L(X, Y ): T(BX)⊂S(BZ), S ∈ A(Z, Y )}, that is, Asur is the smallest surjective ideal containing A. If D⊂Xis a bounded set and UDdenotes the surjection of `1(D) onto Xdefined by UD(ξ) = Px∈Dξ(x)x, then it is easy to show that an operator Tbelongs to Asur(X, Y ) if and only if T◦UBX∈ A(`1(BX), Y ). In the case of a Banach ideal [A, α], Asur becomes a Banach ideal when equipped with the norm αsur(T) = inf{α(S): T(BX)⊂S(BZ), S ∈ A(Z, Y )} =α(T◦UBX). An operator ideal Ais injective if, given S∈ A(X, Z) and T∈ L(X, Y ), the inequality kTxk≤kSxkfor all x∈Ximplies that T∈ A(X, Y ). For an arbitrary ideal A, the injective hull Ainj of Ais the operator ideal with components Ainj(X, Y ) = {T∈ L(X, Y ): kTxk ≤ kSxkfor all x∈X, S ∈ A(X, Z)}, that is, Ainj is the smallest injective ideal containing A. If JYdenotes the canonical embedding of Yinto `∞(BY∗), defined by JY(y)(y∗) = hy∗, yi, then it is easy to show that an operator Tbelongs to Ainj(X, Y ) if and only if JY◦T∈ A(X, `∞(BY∗)). In the case of a Banach ideal [A, α], Ainj becomes 98 J. M. Delgado and C. Pi˜neiro a Banach ideal when equipped with the norm αinj(T) = inf{α(S): kTxk≤kSxkfor all x∈X, S ∈ A(X, Z)} =α(JY◦T). We denote by L,K,Wand Fthe operator ideals of bounded, compact, weakly compact and finite rank linear operators, respectively. We also need the following operator ideals: QNp—quasi p-nuclear operators, Ip— p-integral operators and Πp—p-summing operators. We refer to Pietsch’s book [19] for operator ideals (see also Diestel, Jarchow and Tonge [10] for common operator ideals such as Ipand Πp, and Persson and Pietsch [18] for QNp). 2. A measure of non-A-compactness of a set. Let Abe an operator ideal. A subset Aof the Banach space Xis said to be A-bounded if there exist a Banach space Zand an operator S∈ A(Z, X) with A⊂S(BZ) [22]. The class of A-bounded subsets of Xis denoted by MA(X). Note that an operator belongs to Asur(X, Y ) if and only if it maps bounded subsets of X to A-bounded subsets of Y. The first examples rely on the following fact. Proposition 2.1.A set A⊂Xis A-bounded if and only if UA∈ A(`1(A), X). Proof. If A⊂Xis A-bounded and S∈ A(Z, X) is such that A⊂S(BZ), then UA(B`1(A)) = nX n αnxn:xn∈A, (αn)∈B`1o ⊂nX n αnxn:xn∈S(BZ),(αn)∈B`1o=S(BZ), and it follows that UA(B`1(A)) is A-bounded. Thus, UA∈ Asur(`1(A), X) = A(`1(A), X) [19, Lemma 4.7.3]. The converse is a direct consequence of the inclusion A⊂UA(B`1(A)). Example 2.2.(1) The class of all L-bounded sets in Xcoincides with that of all bounded sets. (2) The class of all K-bounded sets in Xcoincides with that of all relatively compact sets. (3) Let p∈[1,∞). A bounded set A⊂Xis said to be p-limited if for every weakly p-summable sequence (x∗ n) in X∗there exists (αn)∈`psuch that |hx∗ n, xi| ≤ αnfor all x∈Aand n∈N[14]. By [8, Proposition 2.1], A⊂Xis p-limited if and only if U∗ Ais p-summing. So the class of all Πd p-bounded sets in Xis precisely that of all p-limited sets. (4) Let 1 ≤p < ∞and let p0be the conjugate index of p. Denote by Kpthe ideal consisting of all p-compact operators in the sense of Sinha and Duality of measures of non-A-compactness 99 Karn. Since A⊂Xis relatively p-compact if and only if UA∈ Kp(`1(A), X) [9, Proposition 3.5], we deduce that the class of all Kp-bounded sets in Xis precisely that of all relatively p-compact sets. In [4], a special type of A-bounded sets was introduced by Carl and Stephani as a refinement of compactness related to a given operator ideal. A set A⊂Xis said to be A-compact if there exist a Banach space Z, a compact set K⊂Zand an operator S∈ A(Z, X) such that A⊂S(K) (actually, this is the characterization of A-compact sets appearing in [4, Theorem 1.2]). We denote by MA c(X) the class of A-compact subsets of X. Relying on the notion of A-compactness, the notion of A-compact operator is defined in the obvious way: T∈ L(X, Y ) is said to be A-compact if Tmaps bounded sets in Xto relatively A-compact sets in Y. If KAdenotes the class of A-compact operators, then KAis a surjective operator ideal and KA=Asur ◦ K =KA◦ K [4, Theorem 2.1]. From this, it is easy to deduce that A⊂Xis A-compact if and only if UA∈ KA(`1(A), X) and that MA c(X) = MAsur c(X) = MA◦K c(X). Example 2.3.(1) If A=Lor A=K, the class of all A-compact sets in Xcoincides with that of all relatively compact sets. (2) Having in mind the equality Kp=Πd p◦ K (see, for instance, [1, Corollary 4.9]) and the surjectivity of the ideal Πd p(being the dual of an injective ideal), it follows that KΠd p=Kp. So A⊂Xis Πd p-compact if and only if UAis p-compact. By [9, Proposition 3.5], we deduce that the class of all Πd p-compact sets in Xis precisely that of all relatively p-compact sets. (3) Using the above properties, we have MΠd p c(X) = MΠd p◦K c(X) = MKp c(X), that is, the class of all Kp-compact sets in Xis precisely that of all relatively p-compact sets. The notion of A-compactness may be expressed in a similar way to the notion of precompactness in a Banach space. Theorem ([4, Theorem 3.1]).Let [A, α]be a Banach operator ideal, Xa Banach space and A∈MA(X). The following statements are equivalent: (a) Ais A-compact. (b) For every ε > 0, there are finitely many elements x1, . . . , xn∈X, a Banach space Zand an operator S∈ A(Z, X)with α(S)≤εsuch that A⊂ n [ i=1 xi+S(BZ). 100 J. M. Delgado and C. Pi˜neiro The above result is a basis for the following definition of measure of noncompactness referring to a given Banach operator ideal A. Definition 2.4.Let [A, α] be a Banach operator ideal, Xa Banach space and A∈MA(X). The (outer)measure of non-A-compactness of Ais χA(A) = infnε > 0: A⊂ n [ i=1 xi+S(BZ)o, the infimum taken over all possible x1, . . . , xn∈X, Banach spaces Zand operators S∈ A(Z, X) with α(S)≤ε. The condition A∈MA(X) ensures that in the above definition we take the infimum of a nonempty set of positive numbers. Of course, if A⊂B, then χB(·)≤χA(·) and χL≡χ. In this section, we omit the word “outer” when referring to “outer measures of non-A-compactness”. Remark 2.5.It is clear that χA(A) = infnα(S): A⊂ n [ i=1 xi+S(BZ)o, the infimum taken over all possible x1, . . . , xn∈X, Banach spaces Zand operators S∈ A(Z, X). From this, it follows that χA(A) = limnen(A, A), where (en(A, A)) is the sequence of generalized (outer) entropy numbers of the set Awith respect to Aintroduced in [4, Definition 3]. Theorem 3.2 in [4] may be used to obtain the equality χA(A) = χAsur (A) for every A∈ MA(X) = MAsur (X). On the other hand, [7, Proposition 5] shows that χA(A) = inf{α(S): A⊂T(BE) + S(BZ)}, where the infimum is taken over all Banach spaces Eand Zand operators T∈ KA(E, X) and S∈ A(Z, X). Remark 2.6.Taking a glance at Proposition 2.1, it is also possible to conclude that χA(A) = infnε > 0: A⊂ n [ i=1 xi+Bo, the infimum taken over all possible x1, . . . , xn∈Xand A-bounded subsets Bof Xwith α(UB)≤ε. Remark 2.7.In [16], a way to measure the “size” of A-compact sets is introduced as follows. If A⊂Xis A-compact, then one can define mA(A) = inf{α(S): A⊂S(K), S ∈ A(Z, X), K⊂BZcompact}, where the infimum is taken over all Banach spaces Z. It must be pointed out that this notion Duality of measures of non-A-compactness 101 is different from that in Definition 2.4; in fact, a bounded set is A-compact if and only if its mA-measure is finite. Most of the proofs of the following properties are routine, so they are omitted. Proposition 2.8.Assume Ais a Banach operator ideal and A, A1, A2 ⊂Xare A-bounded. Then: (1) χA(A)=0if and only if Ais A-compact. (2) If A1⊂A2, then χA(A1)≤χA(A2). Thus, χA(A1∩A2)≤min{χA(A1), χA(A2)}. (3) χA(A1+A2)≤χA(A1) + χA(A2). As a consequence, χA(∆+A) = χA(A) whenever ∆⊂Xis finite. (4) χA(λA) = |λ|χA(A)for every λ∈R. (5) If T∈ L(X, Y ), then χA(T(A)) ≤ kTkχA(A). (6) If D⊂Xis bounded and T∈ Asur(X, Y ), then χA(T(D)) ≤αsur(T)χ(D), where χ(D)denotes the Hausdorff measure of noncompactness of D. (7) If A2is A-compact, then χA(A1∪A2) = χA(A1). (8) χA(UA(B`1(A))) = χA(A). Proof. (3) Although the idea of the proof is included in [4, Section 4], we give a sketch for completeness. By [4, p. 89, property A], it can be deduced that e2n−1(A1+A2,A)≤en(A1,A) + en(A2,A); hence χA(A1+A2) = lim ne2n−1(A1+A2,A) ≤lim n(en(A1,A) + en(A2,A)) = χA(A1) + χA(A2). (6) If D⊂Xis bounded and T∈ Asur(X, Y ), it is clear that T(D) is Asur-bounded. Let ε>χ(D) and choose x1, . . . , xn∈Xso that D⊂ Sn i=1 xi+εBX. Then T(D)⊂Sn i=1 T(xi)+εT(BX), so in view of Remark 2.5 we have χAsur (T(D)) ≤αsur(εT) = αsur(T)ε. Letting ε&χ(D), we obtain χAsur (T(D)) ≤αsur(T)χ(D), and the property follows since χA≡χAsur [4, Theorem 3.2]. (7) By monotonicity, χA(A1)≤χA(A1∪A2). For the converse inequality, fix ε>χA(A1) so that A1⊂Sn i=1 xi+S1(BZ1) with α(S1)≤ε. Now, for a given δ > 0, the A-compactness of A2ensures the existence of 102 J. M. Delgado and C. Pi˜neiro u1, . . . , um∈Xas well as a Banach space Z2and S2∈ A(Z2, X) with α(S2)≤δsatisfying A2⊂Sm j=1 uj+S2(BZ2). Setting ∆1={x1, . . . , xn}and ∆2={u1, . . . , um}, it is clear that A1∪A2⊂(∆1∪∆2)+S1(BZ1)+S2(BZ2). So, in view of (2), (3) and (6), and having in mind that χ(BE) = 1 whenever Eis infinite-dimensional [3, Theorem 2.5], we conclude that χA(A1∪A2)≤χA(S1(BZ1)) + χA(S2(BZ2)) ≤α(S1)χ(BZ1) + α(S2)χ(BZ2) ≤ε+δ. Letting δ&0 and ε&χA(A1) yields the desired inequality. Remark 2.9.As a consequence of Proposition 2.8(6), every Tin Asur(X, Y ) maps relatively compact subsets of Xto A-compact subsets of Y. For A=Πd p, this means that every operator with p-summing adjoint maps relatively compact subsets to p-compact subsets (as already proved in [9, Theorem 3.14]). It is easy to show that the Hausdorff measure of noncompactness is semiadditive, that is, χ(D1∪D2) = max{χ(D1), χ(D2)}. Apart from the case stated in Proposition 2.8(7), we have not been able to establish whether this property remains true for measures of non-A-compactness with Adifferent from L. In this connection, we have the following result. Proposition 2.10.Let p≥1and let A1, A2⊂Xbe Πd p-bounded sets. Then χΠd p(A1∪A2)≤21/p max{χΠd p(A1), χΠd p(A2)}. Proof. Suppose ε > χΠd p(A1)≥χΠd p(A2) and consider coverings Aj⊂ Snj i=1 xj i+Bjwith πp(UBj)≤ε,j= 1,2 (Remark 2.6). Then A1∪A2⊂[ x∈∆ x+B where ∆={xj i:i= 1, . . . , n1, j = 1, . . . , n2}and B=B1∪B2. It suffices to see that πp(U∗ B)≤21/pε. For any fixed weakly p-summable sequence (x∗ n) in X∗, it is possible to find a partition of Ninto two sets G1and G2such that X n kU∗ Bx∗ nkp≤X n∈G1 kU∗ B1x∗ nkp+X n∈G2 kU∗ B2x∗ nkp. Hence, πp(U∗ B)≤(πp(U∗ B1)p+πp(U∗ B1)p)1/p ≤21/pε. Remark 2.11.If D⊂Xis bounded then χ(D) = χ(D). For an arbitrary Banach operator ideal A, we cannot even ensure that Ais A-bounded Duality of measures of non-A-compactness 103 whenever A⊂Xis. Much more can be said if Aenjoys the following property: Property (P). There exists a positive constant Csuch that, for any Banach spaces Xand Yand T∈ A(X, Y ), we have: (i) T∗∗(BX∗∗ )⊂Y(that is, A⊂W). (ii) The operator e T:BX∗∗ 3x∗∗ 7→ T∗∗x∗∗ ∈Ybelongs to A(X∗∗, Y ). (iii) α(e T)≤Cα(T). Proposition 2.12.Suppose Ais a Banach operator ideal with property (P) and Xis a Banach space. Then: (1) A⊂Xis A-bounded if and only if Ais. (2) χA(A)≤χA(A)≤CχA(A). Proof. Let S∈ A(Z, X) with A⊂S(BZ). Then A⊂e S(BZ∗∗ ). By hypothesis, Sis weakly compact, so it factors through a reflexive Banach space. Thus, e Sis weak∗-weak continuous. From this, e S(BZ∗∗ ) is a weakly compact set in Yand, being absolutely convex, it is norm closed. So we have A⊂e S(BZ∗∗ ), and this shows that Ais A-bounded. Finally, (2) is obtained using a standard argument. If a Banach operator ideal A ⊂ W is regular and satisfies A=Add, then it enjoys property (P). This is the case of operator ideals A ⊂ W and A=Amax [6, pp. 206–207]. Hence, Πd psatisfies property (P) (in fact, Πd p=Kmax p[20, Theorem 12]). Corollary 2.13.If A⊂Xis Πd p-bounded, then χΠd p(A) = χΠd p(A). 3. Measures of non-A-compactness of an operator. If an operator T:X→Yfails to be A-compact, it seems natural to quantify the distance between Tand KA(X, Y ) by evaluating χA(T(BX)) when this expression makes sense. Definition 3.1.Let [A, α] be a Banach operator ideal and let Tbe in Asur(X, Y ). The (outer)measure of non-A-compactness of Tis χA(T) = χA(T(BX)). Note that χA(T) = limnen(T, A) (see [4, Section 4]). When A=L, we are dealing with the so called ball measure of noncompactness. Example 3.2.Let A={en:n∈N} ⊂ c0, where (en) is the unit vector basis in c0. Let us check that χA(A) = 1 if A=Πpor A=Πd p. If Idenotes the embedding map from `1into c0, then ι1(I∗) = 1 (see, for instance, [19, Proposition 6.4.4]), so χId 1(A)≤1. In view of [10, Corollary 5.7], χΠd p(A)≤χΠd 1(A) = χId 1(A)≤1. 110 J. M. Delgado and C. Pi˜neiro On the other hand, X k |hy∗ k,(IdY−PN)yiki|p1/p =X k |h(IdY−PN)∗y∗ k,(IdY−PN)yiki|p1/p ≤X kn X i=1 |h(IdY−PN)∗y∗ k,(IdY−PN)yii|p1/p ≤n X i=1 εp 2pn1/p k((IdY−PN)∗y∗ k)kw p ≤ε 2(1 + λ)k(y∗ k)kw p. Summing up, we have X k |h(T−PN◦T)∗y∗ k, xki|p1/p ≤(1 + λ)(χΠd p(T) + ε)k(y∗ k)kw p, which leads to (4.3). With suitable changes in the preceding result, it is possible to obtain an inequality involving ρΠp(T) and χΠd p(T∗): Theorem 4.2.Let Xand Ybe Banach spaces and 1≤p < ∞. Suppose that X∗has the πλ-approximation property. Then ρΠp(T)≤(1+λ)mΠd p(T∗) for every T∈Πp(X, Y ). We finish with a general version of Theorem 4.1. Theorem 4.3.Let [A, α]be a Banach operator ideal. Let Ybe a Banach space for which there exists a positive constant Lsuch that if E⊂Yis a finite-dimensional space, there exists a finite-dimensional subspace E⊂F⊂ Yand a projection P:Y→Fwith kPk ≤ L. Then ρA(T)≤(1 + L)χA(T) for every T∈ Asur(X, Y ). Proof. Starting as in the proof of Theorem 4.1, set E= span {yi:i= 1, . . . , n}and consider the corresponding subspace Fand the projection P given by the hypothesis. Then the conclusion is a consequence of (T−P◦T)(BX)⊂(IdY−P)(S(BZ)). Acknowledgements. 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