Duality of measures of non-A-compactness

STUDIA MATHEMATICA 229 (2) (2015) Duali y o measu es o non-A-compac ness by Juan Manuel Delgado (Se ille) and C´ andido Pi˜ nei o (Huel a) Abs ac . Le Abe a Banach ope a o ideal. Based on he no ion o A-compac ness in a Banach space due o Ca l and S ephani, we deal wi h he no ion o measu e o non-A-compac ness o an ope a o . We conside a map χA( espec i ely, nA) ac ing on he ope a o s o he su jec i e ( espec i ely, injec i e) hull o Asuch ha χA(T) = 0 ( e- spec i ely, nA(T) = 0) i and only i he ope a o Tis A-compac ( espec i ely, injec i ely A-compac ). Unde ce ain condi ions on he ideal A, we p o e an equi alence inequali y in ol ing χA(T∗) and nAd(T). This inequali y p o ides an ex ension o a p e ious esul s a ing ha an ope a o is quasi p-nuclea i and only i i s adjoin is p-compac in he sense o Sinha and Ka n. 1. In oduc ion. I is well known ha i a bounded subse Ao a Banach space Xis no ela i ely compac , hen he e exis s ε > 0 such ha Acanno be co e ed by ini ely many balls wi h adii smalle han (o equal o) ε. In his se ing, he Hausdo measu e o noncompac ness (o he ball measu e o noncompac ness), χ, is de ined o e e y bounded se A as ollows: χ(A) = in nε > 0: A⊂ n [ i=1 xi+εBXo, whe e BXdeno es he closed uni ball o Xand he in imum is aken o e all possible se s o ini ely many ec o s x1, . . . , xn∈X[11]. O cou se, χ(A) anishes i and only i Ais ela i ely compac . I Tis a (bounded) linea ope a o om he Banach space X o he Banach space Y, he measu e o noncompac ness o Tcan be de ined in a na u al way by se ing χ(T) = χ(T(BX)). Then χis a semino m on L(X, Y ), he space o all bounded linea ope a o s om X o Y, and χ anishes ex- ac ly on K(X, Y ), he subspace o L(X, Y ) consis ing o all compac ope a- 2010 Ma hema ics Subjec Classi ica ion: P ima y 47L20, 47B10; Seconda y 47H08. Key wo ds and ph ases: measu e o noncompac ness, compac se , ope a o ideal, p-summing ope a o , p-compac ope a o , essen ial no m. Recei ed 11 June 2014; e ised 23 No embe 2015. Published online 4 Janua y 2016. DOI: 10.4064/sm7984-1-2016 [95] c Ins y u Ma ema yczny PAN, 2015 96 J. M. Delgado and C. Pi˜nei o o s. Acco ding o Schaude ’s classical heo em, an ope a o T∈ L(X, Y ) is compac i and only i i s adjoin ope a o T∗is. In 1965, Gol’denˇs e˘ın and Ma kus [12] p o ed he inequali ies 1 2χ(T)≤χ(T∗)≤2χ(T), which, in some sense, may be conside ed as an ex ension o Schaude ’s heo- em. Ano he ex ension is ob ained i , o ins ance, he Ku a owski measu e o noncompac ness, γ, is conside ed [15]. The de ini ion o γis simila o ha o χwi h “balls wi h adii” eplaced by “bounded subse s wi h diame e ”. In his case, As ala [2] showed (1.1) γ(T) = γ(T∗) o e e y T∈ L(X, Y ). Based on G o hendieck’s cha ac e iza ion o ela i ely compac se s as hose si ing inside he con ex hull o he no m null sequences, Sinha and Ka n [21] in oduced a s eng hened o m o compac ness in Banach spaces. Le 1 ≤p < ∞and le p0be he conjuga e index o p(i.e., 1/p + 1/p0= 1). A se K⊂Xis said o be ela i ely p-compac i he e exis s a p-summable sequence (xn) in Xsuch ha A⊂ {Pnαnxn: (αn)∈B`p0}((αn)∈Bc0 i p= 1). The no ion o p-compac ope a o is de ined in he ob ious way: an ope a o T∈ L(X, Y ) is said o be p-compac i T(BX) is ela i ely p-compac in Y. Se ano and he p esen au ho s ha e ecen ly p o ed he ollowing: T( espec i ely, T∗) is p-compac i and only i T∗( espec i ely, T) is quasi p-nuclea [9, Co olla y 3.4 and P oposi ion 3.8]. The main pu pose o his pape is o ob ain an ex ension o ha esul using a so o measu es o noncompac ness. Indeed, we conside a posi- i e map χΠd p( espec i ely, nΠp) ac ing on Πd p, he ideal o ope a o s wi h p-summing adjoin s ( espec i ely, Πp, he ideal o p-summing ope a o s) anishing p ecisely on he class o p-compac ope a o s ( espec i ely, quasi p-nuclea ope a o s). Wi h hese maps in hand, an equali y like (1.1) ela - ing χΠd pand nΠpis ob ained (Co olla y 3.13), which p o ides he desi ed gene aliza ion. Ou s udy is ca ied ou in a mo e gene al se ing. Gi en an ope a o ideal A, he no ions o su jec i e ( espec i ely, injec i e) A-compac ness in- oduced in [4] ( espec i ely, [23]) a e basic o his pape . Sec ion 2 is de o ed o he s udy o he map χA, de ined on a ce ain class o bounded subse s o a Banach space ( he so called A-bounded se s), which gi es in o ma ion abou he deg ee o non-A-compac ness o hese se s in such a way ha χA an- ishes p ecisely on he class o (su jec i ely) A-compac se s. In Sec ion 3, he no ion o measu e o non-A-compac ness is ex ended o he ope a o se ing using wo di e en (bu ela ed) app oaches. Indeed, he map χA( espec- Duali y o measu es o non-A-compac ness 97 i ely, nA) gi es in o ma ion abou he deg ee o non-A-compac ness o an ope a o , and i anishes p ecisely on he class o su jec i ely ( espec i ely, injec i ely) A-compac ope a o s. Unde ce ain condi ions on he ideal A, we ob ain se e al inequali ies in ol ing χAand nAac ing on an ope a o and i s adjoin . We show ha his app oach is di e en om ha appea ing in [2] and [24], whe e he no ion o (ou e and inne ) A- a ia ion o an ope - a o is de ined and s udied. Finally, we in oduce he no ion o A-essen ial no m ρAo an ope a o in Sec ion 4 and we s udy he equi alence be ween χAand ρAunde ce ain condi ions on Xo Y. Ou no a ion is s anda d. X,Yand Za e always ese ed o Banach spaces. A Banach space Xwill be ega ded as a subspace o i s bidual X∗∗ unde he canonical embedding iX:X→X∗∗. We deno e he closed uni ball o Xby BX. The Banach space o all bounded linea ope a o s om X o Yis deno ed by L(X, Y ). I Ais an ope a o ideal, hen Addeno es i s dual ope a o ideal, i.e., he one wi h componen s Ad(X, Y ) = {T∈ L(X, Y ): T∗∈ A(Y∗, X∗)}. Recall ha an ope a o ideal Ais su jec i e i , gi en S∈ A(Z, Y ) and T∈ L(X, Y ), he condi ion T(BX)⊂S(BZ) implies ha T∈ A(X, Y ). Fo an a bi a y ideal A, he su jec i e hull Asu o Ais he ope a o ideal whose componen s a e Asu (X, Y ) = {T∈ L(X, Y ): T(BX)⊂S(BZ), S ∈ A(Z, Y )}, ha is, Asu is he smalles su jec i e ideal con aining A. I D⊂Xis a bounded se and UDdeno es he su jec ion o `1(D) on o Xde ined by UD(ξ) = Px∈Dξ(x)x, hen i is easy o show ha an ope a o Tbelongs o Asu (X, Y ) i and only i T◦UBX∈ A(`1(BX), Y ). In he case o a Banach ideal [A, α], Asu becomes a Banach ideal when equipped wi h he no m αsu (T) = in {α(S): T(BX)⊂S(BZ), S ∈ A(Z, Y )} =α(T◦UBX). An ope a o ideal Ais injec i e i , gi en S∈ A(X, Z) and T∈ L(X, Y ), he inequali y kTxk≤kSxk o all x∈Ximplies ha T∈ A(X, Y ). Fo an a bi a y ideal A, he injec i e hull Ainj o Ais he ope a o ideal wi h componen s Ainj(X, Y ) = {T∈ L(X, Y ): kTxk ≤ kSxk o all x∈X, S ∈ A(X, Z)}, ha is, Ainj is he smalles injec i e ideal con aining A. I JYdeno es he canonical embedding o Yin o `∞(BY∗), de ined by JY(y)(y∗) = hy∗, yi, hen i is easy o show ha an ope a o Tbelongs o Ainj(X, Y ) i and only i JY◦T∈ A(X, `∞(BY∗)). In he case o a Banach ideal [A, α], Ainj becomes 98 J. M. Delgado and C. Pi˜nei o a Banach ideal when equipped wi h he no m αinj(T) = in {α(S): kTxk≤kSxk o all x∈X, S ∈ A(X, Z)} =α(JY◦T). We deno e by L,K,Wand F he ope a o ideals o bounded, com- pac , weakly compac and ini e ank linea ope a o s, espec i ely. We also need he ollowing ope a o ideals: QNp—quasi p-nuclea ope a o s, Ip— p-in eg al ope a o s and Πp—p-summing ope a o s. We e e o Pie sch’s book [19] o ope a o ideals (see also Dies el, Ja chow and Tonge [10] o common ope a o ideals such as Ipand Πp, and Pe sson and Pie sch [18] o QNp). 2. A measu e o non-A-compac ness o a se . Le Abe an ope a o ideal. A subse Ao he Banach space Xis said o be A-bounded i he e exis a Banach space Zand an ope a o S∈ A(Z, X) wi h A⊂S(BZ) [22]. The class o A-bounded subse s o Xis deno ed by MA(X). No e ha an ope a o belongs o Asu (X, Y ) i and only i i maps bounded subse s o X o A-bounded subse s o Y. The i s examples ely on he ollowing ac . P oposi ion 2.1.A se A⊂Xis A-bounded i and only i UA∈ A(`1(A), X). P oo . I A⊂Xis A-bounded and S∈ A(Z, X) is such ha A⊂S(BZ), hen UA(B`1(A)) = nX n αnxn:xn∈A, (αn)∈B`1o ⊂nX n αnxn:xn∈S(BZ),(αn)∈B`1o=S(BZ), and i ollows ha UA(B`1(A)) is A-bounded. Thus, UA∈ Asu (`1(A), X) = A(`1(A), X) [19, Lemma 4.7.3]. The con e se is a di ec consequence o he inclusion A⊂UA(B`1(A)). Example 2.2.(1) The class o all L-bounded se s in Xcoincides wi h ha o all bounded se s. (2) The class o all K-bounded se s in Xcoincides wi h ha o all ela- i ely compac se s. (3) Le p∈[1,∞). A bounded se A⊂Xis said o be p-limi ed i o e e y weakly p-summable sequence (x∗ n) in X∗ he e exis s (αn)∈`psuch ha |hx∗ n, xi| ≤ αn o all x∈Aand n∈N[14]. By [8, P oposi ion 2.1], A⊂Xis p-limi ed i and only i U∗ Ais p-summing. So he class o all Πd p-bounded se s in Xis p ecisely ha o all p-limi ed se s. (4) Le 1 ≤p < ∞and le p0be he conjuga e index o p. Deno e by Kp he ideal consis ing o all p-compac ope a o s in he sense o Sinha and Duali y o measu es o non-A-compac ness 99 Ka n. Since A⊂Xis ela i ely p-compac i and only i UA∈ Kp(`1(A), X) [9, P oposi ion 3.5], we deduce ha he class o all Kp-bounded se s in Xis p ecisely ha o all ela i ely p-compac se s. In [4], a special ype o A-bounded se s was in oduced by Ca l and S ephani as a e inemen o compac ness ela ed o a gi en ope a o ideal. A se A⊂Xis said o be A-compac i he e exis a Banach space Z, a compac se K⊂Zand an ope a o S∈ A(Z, X) such ha A⊂S(K) (ac ually, his is he cha ac e iza ion o A-compac se s appea ing in [4, Theo em 1.2]). We deno e by MA c(X) he class o A-compac subse s o X. Relying on he no ion o A-compac ness, he no ion o A-compac op- e a o is de ined in he ob ious way: T∈ L(X, Y ) is said o be A-compac i Tmaps bounded se s in X o ela i ely A-compac se s in Y. I KAde- no es he class o A-compac ope a o s, hen KAis a su jec i e ope a o ideal and KA=Asu ◦ K =KA◦ K [4, Theo em 2.1]. F om his, i is easy o deduce ha A⊂Xis A-compac i and only i UA∈ KA(`1(A), X) and ha MA c(X) = MAsu c(X) = MA◦K c(X). Example 2.3.(1) I A=Lo A=K, he class o all A-compac se s in Xcoincides wi h ha o all ela i ely compac se s. (2) Ha ing in mind he equali y Kp=Πd p◦ K (see, o ins ance, [1, Co olla y 4.9]) and he su jec i i y o he ideal Πd p(being he dual o an injec i e ideal), i ollows ha KΠd p=Kp. So A⊂Xis Πd p-compac i and only i UAis p-compac . By [9, P oposi ion 3.5], we deduce ha he class o all Πd p-compac se s in Xis p ecisely ha o all ela i ely p-compac se s. (3) Using he abo e p ope ies, we ha e MΠd p c(X) = MΠd p◦K c(X) = MKp c(X), ha is, he class o all Kp-compac se s in Xis p ecisely ha o all ela i ely p-compac se s. The no ion o A-compac ness may be exp essed in a simila way o he no ion o p ecompac ness in a Banach space. Theo em ([4, Theo em 3.1]).Le [A, α]be a Banach ope a o ideal, Xa Banach space and A∈MA(X). The ollowing s a emen s a e equi a- len : (a) Ais A-compac . (b) Fo e e y ε > 0, he e a e ini ely many elemen s x1, . . . , xn∈X, a Banach space Zand an ope a o S∈ A(Z, X)wi h α(S)≤εsuch ha A⊂ n [ i=1 xi+S(BZ). 100 J. M. Delgado and C. Pi˜nei o The abo e esul is a basis o he ollowing de ini ion o measu e o noncompac ness e e ing o a gi en Banach ope a o ideal A. De ini ion 2.4.Le [A, α] be a Banach ope a o ideal, Xa Banach space and A∈MA(X). The (ou e )measu e o non-A-compac ness o Ais χA(A) = in nε > 0: A⊂ n [ i=1 xi+S(BZ)o, he in imum aken o e all possible x1, . . . , xn∈X, Banach spaces Zand ope a o s S∈ A(Z, X) wi h α(S)≤ε. The condi ion A∈MA(X) ensu es ha in he abo e de ini ion we ake he in imum o a nonemp y se o posi i e numbe s. O cou se, i A⊂B, hen χB(·)≤χA(·) and χL≡χ. In his sec ion, we omi he wo d “ou e ” when e e ing o “ou e mea- su es o non-A-compac ness”. Rema k 2.5.I is clea ha χA(A) = in nα(S): A⊂ n [ i=1 xi+S(BZ)o, he in imum aken o e all possible x1, . . . , xn∈X, Banach spaces Zand ope a o s S∈ A(Z, X). F om his, i ollows ha χA(A) = limnen(A, A), whe e (en(A, A)) is he sequence o gene alized (ou e ) en opy numbe s o he se Awi h espec o Ain oduced in [4, De ini ion 3]. Theo em 3.2 in [4] may be used o ob ain he equali y χA(A) = χAsu (A) o e e y A∈ MA(X) = MAsu (X). On he o he hand, [7, P oposi ion 5] shows ha χA(A) = in {α(S): A⊂T(BE) + S(BZ)}, whe e he in imum is aken o e all Banach spaces Eand Zand ope a o s T∈ KA(E, X) and S∈ A(Z, X). Rema k 2.6.Taking a glance a P oposi ion 2.1, i is also possible o conclude ha χA(A) = in nε > 0: A⊂ n [ i=1 xi+Bo, he in imum aken o e all possible x1, . . . , xn∈Xand A-bounded subse s Bo Xwi h α(UB)≤ε. Rema k 2.7.In [16], a way o measu e he “size” o A-compac se s is in oduced as ollows. I A⊂Xis A-compac , hen one can de ine mA(A) = in {α(S): A⊂S(K), S ∈ A(Z, X), K⊂BZcompac }, whe e he in imum is aken o e all Banach spaces Z. I mus be poin ed ou ha his no ion Duali y o measu es o non-A-compac ness 101 is di e en om ha in De ini ion 2.4; in ac , a bounded se is A-compac i and only i i s mA-measu e is ini e. Mos o he p oo s o he ollowing p ope ies a e ou ine, so hey a e omi ed. P oposi ion 2.8.Assume Ais a Banach ope a o ideal and A, A1, A2 ⊂Xa e A-bounded. Then: (1) χA(A)=0i and only i Ais A-compac . (2) I A1⊂A2, hen 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) whene e ∆⊂Xis ini e. (4) χA(λA) = |λ|χA(A) o e e y λ∈R. (5) I T∈ L(X, Y ), hen χA(T(A)) ≤ kTkχA(A). (6) I D⊂Xis bounded and T∈ Asu (X, Y ), hen χA(T(D)) ≤αsu (T)χ(D), whe e χ(D)deno es he Hausdo measu e o noncompac ness o D. (7) I A2is A-compac , hen χA(A1∪A2) = χA(A1). (8) χA(UA(B`1(A))) = χA(A). P oo . (3) Al hough he idea o he p oo is included in [4, Sec ion 4], we gi e a ske ch o comple eness. By [4, p. 89, p ope y A], i can be deduced ha 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) I D⊂Xis bounded and T∈ Asu (X, Y ), i is clea ha T(D) is Asu -bounded. Le ε>χ(D) and choose x1, . . . , xn∈Xso ha D⊂ Sn i=1 xi+εBX. Then T(D)⊂Sn i=1 T(xi)+εT(BX), so in iew o Rema k 2.5 we ha e χAsu (T(D)) ≤αsu (εT) = αsu (T)ε. Le ing ε&χ(D), we ob ain χAsu (T(D)) ≤αsu (T)χ(D), and he p ope y ollows since χA≡χAsu [4, Theo em 3.2]. (7) By mono onici y, χA(A1)≤χA(A1∪A2). Fo he con e se inequal- i y, ix ε>χA(A1) so ha A1⊂Sn i=1 xi+S1(BZ1) wi h α(S1)≤ε. Now, o a gi en δ > 0, he A-compac ness o A2ensu es he exis ence o 102 J. M. Delgado and C. Pi˜nei o u1, . . . , um∈Xas well as a Banach space Z2and S2∈ A(Z2, X) wi h α(S2)≤δsa is ying A2⊂Sm j=1 uj+S2(BZ2). Se ing ∆1={x1, . . . , xn}and ∆2={u1, . . . , um}, i is clea ha A1∪A2⊂(∆1∪∆2)+S1(BZ1)+S2(BZ2). So, in iew o (2), (3) and (6), and ha ing in mind ha χ(BE) = 1 whene e Eis in ini e-dimensional [3, Theo em 2.5], we conclude ha χA(A1∪A2)≤χA(S1(BZ1)) + χA(S2(BZ2)) ≤α(S1)χ(BZ1) + α(S2)χ(BZ2) ≤ε+δ. Le ing δ&0 and ε&χA(A1) yields he desi ed inequali y. Rema k 2.9.As a consequence o P oposi ion 2.8(6), e e y Tin Asu (X, Y ) maps ela i ely compac subse s o X o A-compac subse s o Y. Fo A=Πd p, his means ha e e y ope a o wi h p-summing adjoin maps ela i ely compac subse s o p-compac subse s (as al eady p o ed in [9, Theo em 3.14]). I is easy o show ha he Hausdo measu e o noncompac ness is semiaddi i e, ha is, χ(D1∪D2) = max{χ(D1), χ(D2)}. Apa om he case s a ed in P oposi ion 2.8(7), we ha e no been able o es ablish whe he his p ope y emains ue o measu es o non-A-compac ness wi h Adi e en om L. In his connec ion, we ha e he ollowing esul . P oposi ion 2.10.Le p≥1and le A1, A2⊂Xbe Πd p-bounded se s. Then χΠd p(A1∪A2)≤21/p max{χΠd p(A1), χΠd p(A2)}. P oo . Suppose ε > χΠd p(A1)≥χΠd p(A2) and conside co e ings Aj⊂ Snj i=1 xj i+Bjwi h πp(UBj)≤ε,j= 1,2 (Rema k 2.6). Then A1∪A2⊂[ x∈∆ x+B whe e ∆={xj i:i= 1, . . . , n1, j = 1, . . . , n2}and B=B1∪B2. I su ices o see ha πp(U∗ B)≤21/pε. Fo any ixed weakly p-summable sequence (x∗ n) in X∗, i is possible o ind a pa i ion o Nin o wo se s G1and G2such ha 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. Rema k 2.11.I D⊂Xis bounded hen χ(D) = χ(D). Fo an a bi- a y Banach ope a o ideal A, we canno e en ensu e ha Ais A-bounded Duali y o measu es o non-A-compac ness 103 whene e A⊂Xis. Much mo e can be said i Aenjoys he ollowing p op- e y: P ope y (P). The e exis s a posi i e cons an Csuch ha , o any Banach spaces Xand Yand T∈ A(X, Y ), we ha e: (i) T∗∗(BX∗∗ )⊂Y( ha is, A⊂W). (ii) The ope a o e T:BX∗∗ 3x∗∗ 7→ T∗∗x∗∗ ∈Ybelongs o A(X∗∗, Y ). (iii) α(e T)≤Cα(T). P oposi ion 2.12.Suppose Ais a Banach ope a o ideal wi h p op- e y (P) and Xis a Banach space. Then: (1) A⊂Xis A-bounded i and only i Ais. (2) χA(A)≤χA(A)≤CχA(A). P oo . Le S∈ A(Z, X) wi h A⊂S(BZ). Then A⊂e S(BZ∗∗ ). By hypo hesis, Sis weakly compac , so i ac o s h ough a e lexi e Banach space. Thus, e Sis weak∗-weak con inuous. F om his, e S(BZ∗∗ ) is a weakly compac se in Yand, being absolu ely con ex, i is no m closed. So we ha e A⊂e S(BZ∗∗ ), and his shows ha Ais A-bounded. Finally, (2) is ob ained using a s anda d a gumen . I a Banach ope a o ideal A ⊂ W is egula and sa is ies A=Add, hen i enjoys p ope y (P). This is he case o ope a o ideals A ⊂ W and A=Amax [6, pp. 206–207]. Hence, Πd psa is ies p ope y (P) (in ac , Πd p=Kmax p[20, Theo em 12]). Co olla y 2.13.I A⊂Xis Πd p-bounded, hen χΠd p(A) = χΠd p(A). 3. Measu es o non-A-compac ness o an ope a o . I an ope a o T:X→Y ails o be A-compac , i seems na u al o quan i y he dis ance be ween Tand KA(X, Y ) by e alua ing χA(T(BX)) when his exp ession makes sense. De ini ion 3.1.Le [A, α] be a Banach ope a o ideal and le Tbe in Asu (X, Y ). The (ou e )measu e o non-A-compac ness o Tis χA(T) = χA(T(BX)). No e ha χA(T) = limnen(T, A) (see [4, Sec ion 4]). When A=L, we a e dealing wi h he so called ball measu e o noncompac ness. Example 3.2.Le A={en:n∈N} ⊂ c0, whe e (en) is he uni ec o basis in c0. Le us check ha χA(A) = 1 i A=Πpo A=Πd p. I Ideno es he embedding map om `1in o c0, hen ι1(I∗) = 1 (see, o ins ance, [19, P oposi ion 6.4.4]), so χId 1(A)≤1. In iew o [10, Co olla y 5.7], χΠd p(A)≤χΠd 1(A) = χId 1(A)≤1. 110 J. M. Delgado and C. Pi˜nei o On he o he 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 ha e 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 o (4.3). Wi h sui able changes in he p eceding esul , i is possible o ob ain an inequali y in ol ing ρΠp(T) and χΠd p(T∗): Theo em 4.2.Le Xand Ybe Banach spaces and 1≤p < ∞. Suppose ha X∗has he πλ-app oxima ion p ope y. Then ρΠp(T)≤(1+λ)mΠd p(T∗) o e e y T∈Πp(X, Y ). We inish wi h a gene al e sion o Theo em 4.1. Theo em 4.3.Le [A, α]be a Banach ope a o ideal. Le Ybe a Banach space o which he e exis s a posi i e cons an Lsuch ha i E⊂Yis a ini e-dimensional space, he e exis s a ini e-dimensional subspace E⊂F⊂ Yand a p ojec ion P:Y→Fwi h kPk ≤ L. Then ρA(T)≤(1 + L)χA(T) o e e y T∈ Asu (X, Y ). P oo . S a ing as in he p oo o Theo em 4.1, se E= span {yi:i= 1, . . . , n}and conside he co esponding subspace Fand he p ojec ion P gi en by he hypo hesis. Then he conclusion is a consequence o (T−P◦T)(BX)⊂(IdY−P)(S(BZ)). Acknowledgemen s. The au ho s would like o hank P o esso T. Do- m´ınguez-Bena ides o his use ul ideas and commen s while his esea ch was in p ocess. They a e also g a e ul o he e e ee o aluable sugges ions ha imp o ed he pape subs an ially. Duali y o measu es o non-A-compac ness 111 Re e ences [1] K. Ain, R. Lilleme s and E. Oja, Compac ope a o s which a e de ined by `p-spaces, Quaes . Ma h. 35 (2012), 145–159. [2] K. As ala, On measu es o noncompac ness and ideal a ia ions in Banach spaces, Ann. Acad. Sci. Fenn. Se . AI Ma h. Disse a iones 29 (1980), 42 pp. [3] J. M. Aye be-Toledano, T. Dom´ınguez Bena ides and G. L´opez Acedo, Measu es o Noncompac ness in Me ic Fixed Poin Theo y, Ope . Theo y Ad . Appl. 99, Bi kh¨ause , Basel, 1997. [4] B. Ca l and I. S ephani, On A-compac ope a o s, gene alized en opy numbe s and en opy ideals, Ma h. Nach . 119 (1984), 77–95. [5] P. G. Casazza, App oxima ion p ope ies, in: Handbook o he Geome y o Banach Spaces, Vol. 1, No h-Holland, Ams e dam, 2001, 271–316. [6] A. De an and K. Flo e , Tenso No ms and Ope a o Ideals, No h-Holland Ma h. S ud. 176, No h-Holland, Ams e dam, 1993. [7] A. De an and M. S. Ramanujan, Su jec i e A-compac ness and gene alized Kol- mogo o numbe s, No e Ma . 11 (1991), 109–117. [8] J. M. Delgado and C. Pi˜nei o, A no e on p-limi ed se s, J. Ma h. Anal. Appl. 410 (2014), 713–718. [9] J. M. Delgado, C. Pi˜nei o and E. Se ano, Ope a o s whose adjoin s a e quasi p-nuclea , S udia Ma h. 197 (2010), 291–304. [10] J. Dies el, H. Ja chow and A. Tonge, Absolu ely Summing Ope a o s, Camb idge Uni . P ess, Camb idge, 1995. [11] I. Gokhbe g, L. S. Gol’denˇs e˘ın and A. S. Ma kus, In es iga ion o some p ope ies o bounded linea ope a o s in connec ion wi h hei q-no ms, Uch. Zap. Kishine sk. Gos. Uni . 29 (1957), 29–36 (in Russian). [12] L. S. Gol’denˇs e˘ın and A. S. Ma kus, On a measu e o noncompac ness o bounded se s and linea ope a o s, in: S udies in Algeb a and Ma hema ical Analysis, Kishi- ne , 1965, 45–54 (in Russian). [13] H. Ja chow, Locally Con ex Spaces, Teubne , S u ga , 1981. [14] A. K. Ka n and D. P. Sinha, An ope a o summabili y in sequences in Banach spaces, Glasgow Ma h. J. 56 (2014), 427–437. [15] K. Ku a owski, Su les espaces comple s, Fund. Ma h. 15 (1930), 301–309. [16] S. Lassalle and P. Tu co, The Banach ideal o A-compac ope a o s and ela ed app oxima ion p ope ies, J. Func . Anal. 265 (2013), 2452–2464. [17] R. D. Nussbaum, The adius o he essen ial spec um, Duke Ma h. J. 37 (1970), 473–478. [18] A. Pe sson und A. Pie sch, p-nuklea e und p-in eg ale Abbildungen in Banach ¨aumen, S udia Ma h. 33 (1969), 19–62. [19] A. Pie sch, Ope a o Ideals, No h-Holland Ma h. Lib a y 20, No h-Holland, Am- s e dam, 1980. [20] A. Pie sch, The ideal o p-compac ope a o s and i s maximall hull, P oc. Ame . Ma h. Soc. 142 (2014), 519–530. [21] D. P. Sinha and A. K. Ka n, Compac ope a o s whose adjoin s ac o h ough subspaces o `p, S udia Ma h. 150 (2002), 17–33. [22] I. S ephani, Gene a ing sys ems o se s and quo ien s o sujec i e ope a o ideals, Ma h. Nach . 99 (1980), 13–27. [23] I. S ephani, Injec i ely A-compac ope a o s, gene alized inne en opy numbe s and Gel and numbe s, Ma h. Nach . 133 (1987), 247–272. 112 J. M. Delgado and C. Pi˜nei o [24] H.-O. Tylli, The essen ial no m o an ope a o is no sel -dual, Is ael J. Ma h. 91 (1995), 93–110. Juan Manuel Delgado Depa amen o de Ma em´a ica Aplicada I Escuela T´ecnica Supe io de A qui ec u a A enida Reina Me cedes, 2 41012 Se ille, Spain E-mail: [email p o ec ed] C´andido Pi˜nei o Depa amen o de Ma em´a icas Facul ad de Ciencias Expe imen ales Campus Uni e si a io de El Ca men 21071 Huel a, Spain E-mail: [email protected]