On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius
OON THE COMPACT OPERATORS CASE OF THEBISHOP–PHELPS–BOLLOB ´AS PROPERTY FOR NUMERICALRADIUS
DOMINGO GARC´IA, MANUEL MAESTRE, MIGUEL MART´IN, AND ´OSCAR ROLD ´AN
Abstract.
We study the Bishop-Phelps-Bollob´as property for numerical radius re-stricted to the case of compact operators (BPBp-nu for compact operators in short).We show that C p L q spaces have the BPBp-nu for compact operators for every Haus-dorff topological locally compact space L . To this end, on the one hand, we providesome techniques allowing to pass the BPBp-nu for compact operators from subspaces tothe whole space and, on the other hand, we prove some strong approximation propertyof C p L q spaces and their duals. Besides, we also show that real Hilbert spaces andisometric preduals of (cid:96) have the BPBp-nu for compact operators. Introduction, notation, and known results
First we fix some notation in order to be able to describe our aims and results withprecision. Given a Banach space X over the field K of real or complex numbers, wedenote by X ˚ , B X , and S X , its topological dual, its closed unit ball, and its unit sphere,respectively. If Y is another Banach space, L p X, Y q represents the space of all bounded andlinear operators from X to Y , and we denote by K p X, Y q the space of compact operatorsfrom X to Y . When Y “ X , we shall simply write L p X q “ L p X, X q and K p X q “ K p X, X q .Given a locally compact Hausdorff topological space L , C p L q is the Banach space of allscalar-valued continuous functions on L vanishing at infinity.Given an operator T P L p X q , its numerical radius is defined as ν p T q : “ sup t| x ˚ p T p x qq| : p x, x ˚ q P Π p X qu , where Π p X q : “ tp x, x ˚ q P S X ˆ S X ˚ : x ˚ p x q “ u . It is immediate that ν p T q ď } T } forevery T P L p X q and that ν is a seminorm on L p X q . Very often, ν is actually a normon L p X q equivalent to the usual operator norm. The numerical index of the space X measures this fact and it is given by n p X q : “ inf t ν p T q : T P L p X q , } T } “ u“ max t k ě k } T } ď ν p T q , @ T P L p X qu . Date : February 21, 2021.2020
Mathematics Subject Classification.
Primary: 46B04; Secondary: 46B20, 46B25, 46B28.
Key words and phrases.
Banach space; compact operator; Bishop-Phelps-Bollob´as property; numericalradius attaining operator; approximation property.The first and second authors were supported by MINECO and FEDER project MTM2017-83262-C2-1-P and by Prometeo PROMETEO/2017/102. The third author was supported by projects PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta deAnaluc´ıa/FEDER, UE), and FQM-185 (Junta de Andaluc´ıa/FEDER, UE). The fourth author was sup-ported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades, grant FPU17/02023, and byMINECO and FEDER project MTM2017-83262-C2-1-P. a r X i v : . [ m a t h . F A ] F e b D. GARC´IA, M. MAESTRE, M. MART´IN, AND O. ROLD ´AN
It is clear that 0 ď n p X q ď n p X q ą L p X q equivalent to the operator norm. As in this paper we will mainly deal withcompact operators, we will also need the following concept from [11]. Given a Banachspace X , the compact numerical index of X is n K p X q : “ inf t ν p T q : T P K p X q , } T } “ u“ max t k ě k } T } ď ν p T q , @ T P K p X qu . We refer the reader to [11], [18], [19], [21, Subsection 1.1], and references therein for moreinformation and background.An operator S P L p X, Y q is said to attain its norm whenever there exists some x P S X such that } S } “ } S p x q} . An operator T P L p X q is said to attain its numerical radius whenever there exists some p x, x ˚ q P Π p X q such that ν p T q “ | x ˚ p T p x qq| . The sets of normattaining operators from X to Y and of numerical radius attaining operators on X willbe denoted, respectively, by NA p X, Y q and NRA p X q .In 1961, Bishop and Phelps [8] proved that the set NA p X, K q of norm attaining func-tionals on a Banach space X is always dense in X ˚ . However, this result has been shownto fail for general operators between Banach spaces, as Lindenstrauss [22] proved in 1963.We refer the reader to the survey [1] for more information and background on the densityof norm attaining operators, and to [23] for the compact operators version.In 1970, Bollob´as [9] gave a refinement of the Bishop-Phelps Theorem, showing thatyou can approximate simultaneously a functional and a point where it almost attainsits norm by a norm-attaining funcional and a point where the new functional attainsits norm, respectively. In order to extend Bollob´as’ result to norm attaining operatorsbetween Banach spaces, Acosta, Aron, Garc´ıa and Maestre [3] introduced in 2008 theBishop-Phelps-Bollob´as property as follows. Definition 1.1 ([3]) . A pair of Banach spaces p X, Y q has the Bishop-Phelps-Bollob´asproperty ( BPBp for short) if given ε P p , q , there exists η p ε q P p , q such that whenever T P L p X, Y q and x P S X satisfy } T } “ } T p x q} ą ´ η p ε q , there are S P L p X, Y q and x P S X such that } S } “ } S p x q} “ , } x ´ x } ă ε, } S ´ T } ă ε. If the above property holds when we restrict the operators T and S to be compact, wesay that the pair p X, Y q has the Bishop-Phelps-Bollob´as property for compact operators ( BPBp for compact operators for short).With the above notation, the result by Bollob´as just says that the pair p X, K q has theBishop-Phelps-Bollob´as property for every Banach space X . In the paper [3] a variety ofpairs of spaces satisfying the BPBp are provided, together with examples of pairs p X, Y q of Banach spaces failing the BPBp for which NA p X, Y q is dense in L p X, Y q . We refer thereader to the survey [2] and the paper [6] for more information and background on theBPBp.Motivated by this property, Guirao and Kozhushkina [16] introduced in 2013 the Bishop-Phelps-Bollob´as property for numerical radius as follows. Definition 1.2 ([16]) . A Banach space X is said to have the Bishop-Phelps-Bollob´asproperty for numerical radius ( BPBp-nu for short) if for every 0 ă ε ă
1, there exists η p ε q P p , q such that whenever T P L p X q and p x, x ˚ q P Π p X q satisfy ν p T q “ N THE BPBP-NU FOR COMPACT OPERATORS 3 | x ˚ p T p x qq| ą ´ η p ε q , there exist S P L p X q and p y, y ˚ q P Π p X q such that ν p S q “ | y ˚ p S p y qq| “ , } T ´ S } ă ε, } x ´ y } ă ε, } x ˚ ´ y ˚ } ă ε. Since then, several works have been done in order to study what spaces satisfy thatproperty. We summarize next some of the most important results on the matter:(1) The spaces c and (cid:96) have the BPBp-nu [16].(2) L p R q has the BPBp-nu [14].(3) Finite-dimensional spaces have the BPBp-nu [20].(4) The real or complex space L p p µ q has the BPBp-nu for every measure µ when1 ď p ă 8 ([20, Example 8] except for the real case p “
2, which is covered in [21,Corollary 3.3]).(5) Any uniformly convex and uniformly smooth Banach space X with n p X q ą C p K q has the BPBp-nu under some extra conditions on the compactHausdorff space K (for example, when K is metrizable) [7]. Let us comment thatit remains unknown if the result is true for all compact Hausdorff spaces or whathappens in the complex case.We refer the interested reader to the cited papers [7, 14, 16, 20, 21] and the papers[5, 12] and references therein for more information and background.In 2018, Dantas, Garc´ıa, Maestre and Mart´ın [13] studied the BPBp for compact oper-ators. They presented some abstract techniques (based on results about norm attainingcompact operators by Johnson and Wolfe [17]) which allow to carry the BPBp for com-pact operators from sequence spaces (such as c and (cid:96) p ) to function spaces (as C p L q and L p p µ q ). As one of the main results, it is shown in [13] that the BPBp for compact opera-tors of the pair p c , Y q is sufficient to get the BPBp for compact operators of all the pairs p C p L q , Y q regardless of the locally compact Hausdorff topological space L .Our aim in this paper is to study the following property, inspired both by the BPBpfor compact operators and by the BPBp for numerical radius. Definition 1.3.
A Banach space X is said to have the BPBp-nu for compact operators if for every 0 ă ε ă
1, there exists η p ε q P p , q such that whenever T P K p X q and p x, x ˚ q P Π p X q satisfy ν p T q “ | x ˚ p T p x qq| ą ´ η p ε q , there exist S P K p X q and p y, y ˚ q P Π p X q such that ν p S q “ | y ˚ p S p y qq| “ , } T ´ S } ă ε, } x ´ y } ă ε, } x ˚ ´ y ˚ } ă ε. The first work where a somewhat similar property was introduced is [5], where theBPBp-nu for subspaces of L p X q was defined and studied in the case of L p µ q , with µ afinite measure. Let us provide a list of spaces that are known to have the BPBp-nu forcompact operators. Examples 1.4.
The following spaces have the BPBp-nu for compact operators:(a) Finite dimensional spaces [20, Proposition 2].(b) c and (cid:96) (adapting the proofs given in [16, Corollaries 3.3 and 4.2]). D. GARC´IA, M. MAESTRE, M. MART´IN, AND O. ROLD ´AN (c) L p µ q for every measure µ (using [5, Corollary 2.1] for finite measures and adapting[20, Theorem 9] to compact operators for the general case).Adapting the results from [20] and [21], one also has that the L p p µ q spaces have theBPBp-nu for compact operators when 1 ă p ă 8 . However, the adaptation to thecompact operators case of the proofs in [20] and [21] needs to introduce some terminology.Therefore, we enounce the result here but we pospone the proof to Subsection 1.1. Proposition 1.5. L p p µ q has the BPBp-nu for compact operators, for every measure µ and ă p ă 8 . Our main objective in this paper is to prove the following result, which is not coveredby Examples 1.4.
Theorem 1.6. If L is a locally compact Hausdorff space, then C p L q has the BPBp-nufor compact operators. As a consequence, we get that L p µ q spaces have the BPBp-nu for compact operators,completing Example 1.4.c and Proposition 1.5. Corollary 1.7. L p µ q has the BPBp-nu for compact operators for every measure µ . Let us recall that it is shown in [7] that the real space C p K q has the BPBp-nu for somecompact Hausdorff spaces K , but the general case, as well as the complex case, remainopen as far as we know. However, Theorem 1.6 gives a complete answer in the case ofcompact operators.To get the proof of Theorem 1.6, we need two kind of ingredients. On the one hand,we provide in Section 2 some abstract results that will allow us to carry the BPBp-nu forcompact operators from sequence spaces into function spaces, in some cases. The moregeneral result of this kind is Lemma 2.1, which will be the first ingredient for the proof ofTheorem 1.6. It is somehow an extension of [13, Lemma 2.1] but it needs more restrictivehypothesis in order to deal with the numerical radius instead of with the norm of theoperators. We also provide in that section some useful particular cases of Lemma 2.1which allow to show, for instance, that every predual of (cid:96) has the BPBp-nu for compactoperators, see Corollary 2.6. The second ingredient for the proof of Theorem 1.6 is somestrong approximation property of C p L q spaces and their duals which will be provided inSection 3 (see Theorem 3.4) and which will allow us to apply Lemma 2.1 in this case, thusproviding the proof of Theorem 1.6. Let us also comment that Theorem 3.4 gives a muchstronger approximation property of C p L q and its dual space than [13, Lemma 3.4].1.1. L p spaces have the BPBp-nu for compact operators. In this subsection, we will adapt the concepts and results from [20] and [21] to thecompact operators case to show that L p p µ q spaces have the BPBp-nu for compact operatorsfor 1 ă p ă 8 , providing thus a proof of Proposition 1.5.In [20, Definition 5] a weaker version of the BPBp-nu, the weak BPBp-nu, was intro-duced and studied. We present here the compact operators version of that property. Definition 1.8.
A Banach space X is said to have the weak BPBp-nu for compact oper-ators if for every 0 ă ε ă
1, there exists η p ε q P p , q such that whenever T P K p X q and N THE BPBP-NU FOR COMPACT OPERATORS 5 p x, x ˚ q P Π p X q satisfy ν p T q “ | x ˚ p T p x qq| ą ´ η p ε q , there exist S P K p X q and p y, y ˚ q P Π p X q such that ν p S q “ | y ˚ p S p y qq| , } T ´ S } ă ε, } x ´ y } ă ε, } x ˚ ´ y ˚ } ă ε. Note that this is a similar property to the BPBp-nu for compact operators, but withoutasking for the condition ν p S q “ Proposition 1.9.
If a Banach space is uniformly convex and uniformly smooth, then ithas the weak BPBp-nu for compact operators.Proof.
We can follow the proof of [20, Proposition 4], just keeping in mind that if theoriginal operator T is compact, then the rest of operators T n from that proof are alsocompact, and so, S is compact too. (cid:3) Later, in [20, Proposition 6], it is proven that in Banach spaces with positive numericalindex, the BPBp-nu and the weak BPBp-nu are equivalent. This property is also true forthe compact operators versions of the properties if we use the compact numerical index.
Proposition 1.10.
Let X be a Banach space such that n K p X q ą . Then X has theBPBp-nu if, and only if, it has the weak BPBp-nu.Proof. It suffices to follow the proof from [20, Proposition 6] but with both T and S beingnow compact operators, and using n K p X q instead of n p X q . (cid:3) As a consequence of these two results, similarly to what is done in [20], we get that all L p p µ q spaces have the BPBp-nu for compact operators when 1 ă p ă 8 in the complexcase and when 1 ă p ă 8 , p ‰ n K p L p p µ qq ě n p L p p µ qq ą ` ă p ă 8 , p ‰ ˘ by [24] and, on the other hand, n K p X q ě { e ą p in p , `8q in the complexcase and for all values of p in p , `8q except for p “ X , we consider the following subset of K p X q : Z K p X q : “ (cid:32) T P K p X q : ν p T q “ ( which is the set of all skew-hermitian compact operators on X . Observe that Z K p X q “ K p X q X Z p X q , where Z p X q is the Lie-algebra of all skew-hermitian operators on X (see [21, p. 1004] forinstance). Adapting the concept of second numerical index given in [21], we define the D. GARC´IA, M. MAESTRE, M. MART´IN, AND O. ROLD ´AN second numerical index for compact operators of a Banach space X as the constant n K p X q : “ inf (cid:32) ν p T q : T P K p X q , } T ` Z K p X q} “ ( “ max (cid:32) M ě M } T ` Z K p X q} ď ν p T q for all T P K p X q ( , where } T ` Z K p X q} is the quotient norm in K p X q{ Z K p X q .The next result is a version for compact operators of [21, Theorem 3.2]. Proposition 1.11.
Let X be a real Banach space with n K p X q ą . Then, the BPBp-nufor compact operators and the weak BPBp-nu for compact operators are equivalent in X .Proof. It suffices to adapt the steps from the proof of [21, Theorem 3.2] to the case ofcompact operators. That is: all the involved operators T , S , S and S are now compact,the Z p X q set is replaced by Z K p X q , and the index n p X q is replaced by n K p X q . (cid:3) We are going to see next that the second numerical index for compact operators of areal Hilbert space equals one.
Proposition 1.12.
Let H be a real Hilbert space. Then, n K p H q “ . The proof of this result will be an adaptation of the one of [21, Theorem 2.3]. Recallthat in a real Hilbert space endowed with an inner product p¨|¨q , H ˚ identifies with H bythe isometric isomorphism x ÞÝÑ p¨| x q . Therefore, Π p X q “ tp x, x q P H ˆ H : x P S H u ,and so, for every T P L p H q , one has ν p T q “ sup t|p T x | x q| : x P S H u . We first need to givethe compact operators version of [21, Lemma 2.4] whose proof is an obvious adaptationof the proof of that result. Lemma 1.13.
Let H be a real Hilbert space.(a) Z K p H q “ t T P K p H q : T “ ´ T ˚ u .(b) If T P K p H q is selfadjoint (i.e. T “ T ˚ ), then } T } “ ν p T q . We are now ready to present the pending proof of Proposition 1.12.
Proof of Proposition 1.12.
It suffices to adapt the proof of [21, Theorem 2.3] to the com-pact operators case, that is: the involved operators T and S are now compact, and theset Z p X q is replaced by Z K p X q . (cid:3) As a consequence of Propositions 1.9, 1.11, and 1.12, we get the following result whichprovides the proof of the pending part of Proposition 1.5.
Corollary 1.14. If H is a real Hilbert space, then it has the BPBp-nu for compact oper-ators. First ingredient: the tools
In this section, we will provide an abstract result that will allow us later to carrythe BPBp-nu for compact operators from some sequence spaces to function spaces. Themost general version that we are able to prove is the following, which is inspired in [13,Lemma 2.1], but it needs more requirements. We need some notation first. An absolutenorm | ¨ | a is a norm in R such that |p , q| a “ |p , q| a “ |p s, t q| a “ |p| s | , | t |q| a forevery p s, t q P R . Given a Banach space X , we say that a projection P on X is an absoluteprojection if there is an absolute norm | ¨ | a such that } x } “ ˇˇ p} P p x q} , } x ´ P p x q}q ˇˇ a for N THE BPBP-NU FOR COMPACT OPERATORS 7 every x P X . Examples of absolute projections are the M - and L -projections and, morein general, the (cid:96) p -projections. We refer the reader to [13] for the use of absolute normswith the Bishop-Phelps-Bollob´as type properties and to the references therein for moreinformation on absolute norms. Lemma 2.1.
Let X be a Banach space satisfying that n K p X q ą . Suppose that there isa mapping η : p , q ÝÑ p , q such that given δ ą , x ˚ , . . . , x ˚ n P B X ˚ and x , . . . , x (cid:96) P B X , we can find norm one operators r P : X ÝÑ r P p X q , i : r P p X q ÝÑ X such that for P : “ i ˝ r P : X ÝÑ X , the following conditions are satisfied:(i) } P ˚ p x ˚ j q ´ x ˚ j } ă δ , for j “ , . . . , n .(ii) } P p x j q ´ x j } ă δ , for j “ , . . . , (cid:96) .(iii) r P ˝ i “ Id r P p X q .(iv) r P p X q satisfies the Bishop-Phelps-Bollob´as property for numerical radius for com-pact operators with the mapping η .(v) Either P is an absolute projection and i is the natural inclusion, or n K p r P p X qq “ n K p X q “ .Then, X satisfies the BPBp-nu for compact operators. Let us comment on the differences between the lemma above and [13, Lemma 2.1].First, condition (ii) is more restrictive here than in that lemma, where it only dealt withone point. Second, the requirements of item (v) on the compact numerical index or on theabsoluteness of the projections did not appear in [13, Lemma 2.1], but they are neededhere as numerical radius does not behave well in general with respect to extensions ofoperators.
Proof.
Given ε P p , q , let ε p ε q be the unique number with 0 ă ε p ε q ă ε p ε q ˆ ` p ´ ε p ε qq n K p X q ˙ “ ε, which, in particular, satisfies that ε p ε q ă ε . From now on, we shall simply write ε insteadof ε p ε q . We define next(1) η p ε q : “ min ε p n K p X qq , ` η ` ε ˘˘ p n K p X qq + ` ε P p , q ˘ , where η is the function appearing in the hypotheses of the lemma. We fix T P K p X q with ν p T q “ } T } ď n K p X q ) and p x , x ˚ q P Π p X q such that | x ˚ p T p x qq| ą ´ η p ε q . Since T ˚ p B X ˚ q is relatively compact, we can find x ˚ , . . . , x ˚ n P B X ˚ such thatmin ď j ď n } T ˚ p x ˚ q ´ x ˚ j } ă η p ε q for all x ˚ P B X ˚ . Similarly, since T p B X q is relatively compact, we can find x , . . . , x (cid:96) P B X such thatmin ď j ď (cid:96) } T p x q ´ x j } ă η p ε q for all x P B X . Let r P : X ÝÑ r P p X q , i : r P p X q ÝÑ X and P : “ i ˝ r P : X ÝÑ X satisfying the conditions(i)-(v) for x , . . . , x (cid:96) P B X , x ˚ , . . . , x ˚ n P B X ˚ and δ “ η p ε q . D. GARC´IA, M. MAESTRE, M. MART´IN, AND O. ROLD ´AN
Now, for every x ˚ P B X ˚ , we have } T ˚ p x ˚ q ´ P ˚ p T ˚ p x ˚ qq}ď min ď j ď n (cid:32) } T ˚ p x ˚ q ´ x ˚ j } ` } x ˚ j ´ P ˚ p x ˚ j q} ` } P ˚ p x ˚ j q ´ P ˚ p T ˚ p x ˚ qq} ( ă η p ε q , and hence, } T ´ T P } “ } T ˚ ´ P ˚ T ˚ } ď η p ε q . On the other hand, for each x P B X , wehave } T p x q ´ P p T p x qq} ď min ď j ď (cid:96) t} T p x q ´ x j } ` } x j ´ P p x j q} ` } P p x j q ´ P p T p x qq}uă η p ε q , and then, } T ´ P T } ď η p ε q . Therefore, } P T P ´ T } ď } P T P ´ P T } ` }
P T ´ T } ď } T P ´ T } ` } P T ´ T } ď η p ε q . Consider p r P p x q , i ˚ p x ˚ qq P r P p X q ˆ p r P p X qq ˚ . Note that it is not true in general that p r P p x q , i ˚ p x ˚ qq P Π p r P p X qq , but we have that } r P p x q} ď } i ˚ p x ˚ q} ď
1, and also, that x ˚ p i p r P p x qqq “ x ˚ p x q loomoon “ ´ x ˚ p i p r P p x qq ´ x q loooooooooomoooooooooon } P x ´ x }ă η p ε q ùñ Re p x ˚ p i p r P p x qqqq ě ´ η p ε q . By the Bishop-Phelps-Bollob´as Theorem (see [10, Corollary 2.4.b] for this version), thereexist p y, y ˚ q P Π p r P p X qq satisfying thatmax ! } y ´ r P p x q} , } y ˚ ´ i ˚ p x ˚ q} ) ď a η p ε q ď ε . Next, we observe that the following two inequalities hold: } r P ˚ p y ˚ q ´ x ˚ } ď } r P ˚ p y ˚ q ´ r P ˚ p i ˚ p x ˚ qq} ` } r P ˚ p i ˚ p x ˚ qq ´ x ˚ }ď a η p ε q ` η p ε q ď ε . (2)(3) } i p y q ´ x } ď } i p y q ´ i p r P p x qq} ` } i p r P p x qq ´ x } ď a η p ε q ` η p ε q ď ε . Let T : “ r P ˝ T ˝ i : r P p X q ÝÑ r P p X q . Claim.
We have that | y ˚ p T y q| ą ´ η ´ ε ¯ and | y ˚ p T y q| ą ´ ε . Indeed, from equations (2) and (3), we obtain that ˇˇ x ˚ p T p x qq´ r P ˚ p y ˚ p T p i p y qqqq ˇˇ ď | x ˚ p T p x qq ´ x ˚ p T p i p y qqq| ` | x ˚ p T p i p y qqq ´ r P ˚ p y ˚ p T p i p y qqqq|ď } T }} x ´ i p y q} ` } T }} x ˚ ´ r P ˚ p y ˚ q}ď } T } ´a η p ε q ` η p ε q ¯ . Now, we can estimate | y ˚ p T p y qq| as follows: | y ˚ p T p y qq| “ ˇˇ r P ˚ p y ˚ p T p i p y qqqq ˇˇ ě ˇˇ x ˚ p T p x qq ˇˇ ´ ˇˇ x ˚ p T p x qq ´ r P ˚ p y ˚ p T p i p y qqqq ˇˇ ě ´ η p ε q ´ } T } a η p ε q ´ } T } η p ε q . N THE BPBP-NU FOR COMPACT OPERATORS 9
From here, using the definition of η p ε q given in Eq. (1) and the fact that } T } ď { n K p X q ,we get both assertions of the claim.In particular, we get that ν p T q ě ´ ε ą
0. On the other hand, we also have that ν p T q ď
1. Indeed, if there were some p q, q ˚ q P Π p r P p X qq with | q ˚ p T p q qq| ą
1, we wouldget | q ˚ p T p q qq| “ | q ˚ p r P p T p i p q qqqq| “ |p r P ˚ p q ˚ qqp T p i p q qqq| ą , but ν p T q “
1, and p r P ˚ p q ˚ qqp i p q qq “ q ˚ p r P p i p q qqq “ q ˚ p q q “ . Thus p i p q q , r P ˚ p q ˚ qq P Π p X q , and that is a contradiction.We define now the operator r T : “ T ν p T q . Clearly, r T is a compact operator such that ν p r T q “
1. From the claim, we get that ˇˇ y ˚ p r T p y qq ˇˇ “ ν p T q | y ˚ p T p y qq| ě | y ˚ p T p y qq| ą ´ η ´ ε ¯ . Now, since r P p X q has the BPBp-nu for compact operators with the mapping η , there exista compact operator r S : r P p X q ÝÑ r P p X q with ν p r S q “ p z, z ˚ q P Π p r P p X qq such that ν p r S q “ ˇˇ z ˚ p r S p z qq ˇˇ “ , } z ´ y } ă ε , } z ˚ ´ y ˚ } ă ε , } r S ´ r T } ă ε . Let t “ i p z q P B X and t ˚ “ r P ˚ p z ˚ q P B X ˚ . We have that t ˚ p t q “ z ˚ p r P p i p z qqq “ z ˚ p z q “ . Thus p t, t ˚ q P Π p X q , and also, by (2) and (3), } t ´ x } ď } t ´ i p y q} ` } i p y q ´ x } “ } i p z q ´ i p y q} ` } i p y q ´ x } ă ε ` ε “ ε ď ε, } t ˚ ´ x ˚ } ď } r P ˚ p z ˚ q ´ r P ˚ p y ˚ q} ` } r P ˚ p y ˚ q ´ x ˚ } ă ε ` ε “ ε ď ε. We define S “ i ˝ r S ˝ r P : X ÝÑ X , which is a compact operator. It is clear that ν p S q ě | t ˚ p S p t qq| “ | z ˚ p r P p i p r S p r P p i p z qqqqqq| “ | z ˚ p r S p z qq| “ . Also, } S ´ T } “ } i ˝ r S ˝ r P ´ T }ď } i ˝ r S ˝ r P ´ i ˝ r T ˝ r P } ` } i ˝ r T ˝ r P ´ P T P } ` }
P T P ´ T }“ } i ˝ r S ˝ r P ´ i ˝ r T ˝ r P } ` ›››› P T Pν p T q ´ P T P ›››› ` }
P T P ´ T }ď } r S ´ r T } ` } T } ¨ ˇˇˇˇ ν p T q ´ ˇˇˇˇ ` } P T P ´ T } and, since } T } ď n K p X q , 1 ´ ε ď ν p T q ď
1, and 6 η p ε q ď ε , we continue as: ď ε ` ε p ´ ε q n K p X q ` η p ε q ď ε ˆ ` p ´ ε q n K p X q ˙ ă ε. We finish the proof if we prove that ν p S q ď
1. We consider the following cases: ‚ Case 1: if r P is an absolute projection and i is the natural inclusion, as a conse-quence of [11, Lemma 3.3], we get that ν p S q “ ν p i ˝ r S ˝ r P q “ ν p r S q “ . ‚ Case 2: if n K p X q “ n K p r P p X qq “
1, then ν p S q “ } S } ď } r S } “ ν p r S q “ . Hence, the result follows in the two cases. (cid:3)
We will now provide some applications and consequences of the previous lemma. Givena continuous projection P : X ÝÑ X , if we set r P : X ÝÑ r P p X q “ P p X q Ă X (that is, r P is just the operator P with a restricted codomain) and i : P p X q ÝÑ X is the naturalinclusion then, trivially, we have that P “ i ˝ r P and that r P ˝ i “ Id r P p X q . This easyobservation allows to get the following particular case of Lemma 2.1. Proposition 2.2.
Let X be a Banach space with n K p X q ą . Suppose that there exists anet t P α u α P Λ of norm-one projections on X satisfying that t P α p x qu ÝÑ x for all x P X and t P ˚ α p x ˚ qu ÝÑ x ˚ for all x ˚ P X ˚ , and that there exists a function η : p , q ÝÑ p , q suchthat all the spaces P α p X q with α P Λ have the BPBp-nu for compact operators with thefunction η . Suppose, moreover, that for each α P Λ , at least one of the following conditionsis satisfied:(1) the projection P α is absolute,(2) n K p P α p X qq “ n K p X q “ .Then, the space X has the BPBp-nu for compact operators. We may now obtain the following consequence of the above result. Given a Banachspace X and m P N , the space (cid:96) m p X q represents the (cid:96) -sum of m copies of X , and we willwrite (cid:96) p X q for the (cid:96) -sum of countably infinitely many copies of X . Similarly, c p X q isthe c -sum of countably infinitely many copies of X . When X “ K , we just write (cid:96) m for (cid:96) m p K q . Corollary 2.3.
Let X be a Banach space with n K p X q ą . Then, the following statementsare equivalent:(i) The space c p X q has the BPBp-nu for compact operators.(ii) There is a function η : p , q ÝÑ p , q such that all the spaces (cid:96) n p X q , with n P N ,have the BPBp-nu for compact operators with the function η .Moreover, if X is finite dimensional, these properties hold whenever c p X q or (cid:96) p X q havethe BPBp-nu.Proof. That (ii) implies (i) is a consequence of Proposition 2.2 since for every n P N , theoperator on c p X q which is the identity on the first n coordinates and 0 elsewhere is anabsolute projection whose image is isometrically isomorphic to (cid:96) n p X q .(i) implies (ii) is a consequence of [12, Proposition 4.3], as one can easily see (cid:96) n p X q as an (cid:96) -summand of c p X q . Let us comment that the function η valid for all (cid:96) n p X q isthe function valid for c p X q and this actually follows from the proof of [12, Theorem 4.1](from which [12, Proposition 4.3] actually follows). N THE BPBP-NU FOR COMPACT OPERATORS 11
Finally, when X has finite dimension, if c p X q or (cid:96) p X q has the BPBp-nu, then con-dition (ii) holds by using [12, Theorem 4.1] and the fact that (cid:96) n p X q is finite-dimensionaland so, every operator from (cid:96) n p X q to itself is compact. (cid:3) As stated in Examples 1.4, that c and the spaces (cid:96) n for n P N have the BPBp-nu forcompact operators is a consequence of [16, Corollary 4.2] and [20, Proposition 2]. Actually,the fact that all the space (cid:96) n have the BPBp-nu with the same function η follows from[16, Corollary 4.2] and (the proof of) [12, Theorem 4.1]. However, let us note that we canalso get this result as a consequence of our previous corollary. Corollary 2.4.
There is a function η : p , q ÝÑ p , q such that the space c and thespaces (cid:96) n with n P N , have the BPBp-nu for compact operators with the function η . Additionally, [12, Proposition 4.3] also implies that whenever (cid:96) n p X q has the BPBp-nufor compact operators for some n P N , then so does X , although the converse remainsunknown in general (even for n “ Corollary 2.5.
Let X be a Banach space with n K p X q ą . Suppose that there exists anet t P α u α P Λ of norm-one projections on X such that α ĺ β implies P α p X q Ă P β p X q , that t P ˚ α p x ˚ qu ÝÑ x ˚ for all x ˚ P X ˚ , and that there exists a function η : p , q ÝÑ p , q suchthat all the spaces P α p X q with α P Λ have the BPBp-nu for compact operators with thefunction η . Suppose, moreover, that for each α P Λ , at least one of the following conditionsis satisfied:(1) the projection P α is absolute,(2) n K p P α p X qq “ n K p X q “ .Then, the space X has the BPBp-nu for compact operators.Proof. Observe that in order to apply Proposition 2.2 we only need that t P α x u ÝÑ x innorm for all x P X . But this is proved in [13, Corollary 2.4], so we are done. (cid:3) The previous result can be used to prove that all the preduals of (cid:96) have the BPBp-nufor compact operators. Corollary 2.6.
Let X be a Banach space such that X ˚ is isometrically isomorphic to (cid:96) .Then X has the BPBp-nu for compact operators.Proof. By using a deep result due to Gasparis [15], it is shown in the proof of [13, Theorem3.6] that there exists a sequence of norm-one projections P n : X ÝÑ X satisfiying that P n ` P n “ P n (and so, P n p X q Ă P n ` p X q ), that P n p X q is isometrically isomorphic to (cid:96) n ,and also that P ˚ n p x ˚ q ÝÑ x ˚ for all x ˚ P X ˚ (this claim holds since the sets Y n defined onthat proof satisfy that their union is dense in X ˚ “ (cid:96) ).Next, as P n p X q is isometrically isomorphic to (cid:96) n , on the one hand we have that allthe spaces P n p X q have the BPBp-nu for compact operators with the same function η asa consequence of Corollary 2.4. On the other hand, n p X q “ n p P n p X qq “ n P N (see [18], for instance) so, in particular, n K p X q “ n K p P n p X qq “ n P N . Finally,Corollary 2.5 provide the desired result. (cid:3) Second ingredient: a strong approximation property of C p L q spaces andtheir duals The aim of this section is to provide some strong approximation property of C p L q spaces and their duals which allow to use Lemma 2.1 (actually, Proposition 2.2) to give aproof of Theorem 1.6. We need a number of technical lemmas. Lemma 3.1.
Let L be a locally compact space, let t K , . . . , K M u be a family of pairwisedisjoint non-empty compact subsets of L , and let K Ă L be a compact set with M Ť m “ K m Ă K . If t U , . . . , U R u is a family of relatively compact open subsets of L covering K such thatfor each m there is an r p m q with K m Ă U r p m q , m “ , . . . , M , then there exists an openrefinement t Z , . . . , Z S u , M ď S ď R ` M with Z , . . . , Z M pairwise disjoint, satisfying:(1) For m “ , . . . , M , K m Ă Z m , and K m X Z s “ H for all s P t , . . . , S uzt m u .(2) For all s ą M , there exists z s P Z s z ˆ Ť s ‰ s Z s ˙ .Proof. As t K , . . . , K M u are pairwise disjoint, there exist t V , . . . V M u pairwise disjointopen subsets of L with K m Ă V m Ă U r p m q , m “ , . . . , M .The family " V , . . . , V M , U z ˆ M Ť m “ K m ˙ , . . . , U R z ˆ M Ť m “ K m ˙* is another cover of K by open subsets of L subordinated to t U r u Rr “ . We define Z m : “ V m for m “ , . . . , M ,and W r : “ U r z ˆ M Ť m “ K m ˙ for r “ , . . . , R .If W Ă V Y . . . Y V M , then t V , . . . , V M , W , . . . , W R u is again a cover of K . If thathappens again and again until W R , we have that t Z , . . . , Z M u is the cover we were lookingfor. In other case, let r ě w r P W r z ˆ M Ť m “ V m ˙ , and denote Z M ` : “ W r . The family t V , . . . , V M , W r , W r ` , . . . , W R u is a cover of K by open sets, and then, so is the family (cid:32) V , . . . , V M , W r , W r ` zt w r u , . . . , W R zt w r u ( . Consider now r ą r the first natural number such that there exists w r P W r zt w r u and w r R V Y . . . Y V M Y W r . Let Z M ` : “ W r zt w r u and proceed as before. In at most R steps, we get t Z , . . . , Z S u , M ď S ď R ` M , such that ‚ K m Ă Z m for m “ , . . . M . ‚ ˆ M Ť m “ K m ˙ X Z s “ H for s ą M . ‚ For all s ą M , there exists w r s ´ M P Z s z ˆ Ť s ‰ s Z s ˙ . (cid:3) We next provide a result showing the existence of certain partitions of the unity. Weseparate the non-compact case (Lemma 3.2) and the compact case (Lemma 3.3) for thesake of clarity. We start with the non-compact case.
Lemma 3.2.
Let L be a non-compact locally compact space. Let K Ă L be a compact setand t K , . . . , K M u a family of pairwise disjoint non-empty compact subsets of K . Given a N THE BPBP-NU FOR COMPACT OPERATORS 13 family t U , . . . , U R u of relatively compact open subsets of L that cover K , let t Z , . . . , Z S u be a family of open subsets of L covering K such that they satisfy the thesis of Lemma 3.1,and denote by Z S ` the set L z ˆ S Ť s “ Z s ˙ . Then, there exists a partition of the unitysubordinated to t Z s u S ` s “ , t ϕ s u S ` s “ , such that:(1) t ϕ , . . . , ϕ M u have disjoint support.(2) ϕ m p K m q ” , for m “ , . . . , M .(3) For all M ă s ď S ` , there exists z s P Z s such that ϕ s p z s q “ .(4) For s “ , . . . , S ` , supp p ϕ s q Ă Z s .(5) p ϕ ` ¨ ¨ ¨ ` ϕ S qp x q “ , for all x P K .Proof. By hypothesis, there exists some z S ` P L z ˆ S Ť s “ Z s ˙ , since for all s , Z s Ă R Ť r “ U r ,which is a compact set. Now, we follow the argument from the proof of [25, Theorem2.13], but adapted to our case.As K Ă Z Y . . . Y Z S , for each x P K , there exists a neighbourhood of x , Y x , withcompact closure Y x Ă Z s for some s . Consider x , . . . , x p such that K Ă Y x Y . . . Y Y x p .For each 1 ď s ď S , let H s be the union of those Y x j which lie in Z s , and if M ă s ď S ,we take H s Y t z s u , with z s P Z s z ˆ Ť s ‰ s Z s ˙ . Note that the sets H , . . . , H M and H M ` Yt z M ` u , . . . , H S Yt z S u are non-empty. By Urysohn’s Lemma, there are continuousfunctions g s : L ÝÑ r , s such that g s p H s q ” g s ˇˇ L z Z s ”
0, for 1 ď s ď M , and g s p H s Y t z uq ” g s ˇˇ L z Z s ” M ă s ď S . Define ϕ : “ g ,ϕ : “ p ´ g q g , ... ϕ S : “ p ´ g qp ´ g q ¨ ¨ ¨ p ´ g S ´ q g S Clearly, supp p ϕ s q Ă Z s for all s “ , . . . , S , and we have that ϕ ` ¨ ¨ ¨ ` ϕ S “ ´ p ´ g q ¨ ¨ ¨ p ´ g S q . Since K Ă H Y . . . Y H S , for each x P K , there exists s “ s p x q with g s p x q “
1, and also,for all s “ , . . . , M , we have that t x P L : ϕ s p x q ‰ u Ă t x P L : g s p x q ‰ u Ă Z s . Therefore, the functions t ϕ , . . . , ϕ M u have disjoint support, and ϕ ` ¨ ¨ ¨ ` ϕ S ” K .We define ϕ S ` : “ ´ p ϕ ` ¨ ¨ ¨ ` ϕ S q “ p ´ g q ¨ ¨ ¨ p ´ g S q . Moreover, K m Ă Z m for m “ , . . . , M , and K m X Z s “ H for m ‰ s , m “ , . . . , M , s “ , . . . , S . Hence, ϕ m p x q “ S ÿ s “ ϕ s p x q “ , @ x P K m , m “ , . . . , M. On the other hand, if M ă s ď S , let z s P Z s z ˆ Ť s ‰ s Z s ˙ . We have that ϕ s p z s q “ S ÿ s “ ϕ s p z s q “ , and z S ` R S ď s “ Z s , thus ϕ S ` p z S ` q “ . (cid:3) The next result is the version of the previous lemma for compact topological spaces.
Lemma 3.3.
Let L be a compact space. Let t K , . . . , K M u be a family of pairwise disjointnon-empty compact subsets of L . Given a family t U , . . . , U R u of relatively compact opensubsets of L that cover it, let t Z , . . . , Z S u be a family of open subsets of L covering K such that they satisfy the thesis of Lemma 3.1. Then, there exists a partition of the unitysubordinated to t Z s u Ss “ , t ϕ s u Ss “ , such that:(1) t ϕ , . . . , ϕ M u have disjoint support.(2) ϕ m p K m q ” , for m “ , . . . , M .(3) For all M ă s ď S , there exists z s P Z s such that ϕ s p z s q “ .(4) For s “ , . . . , S , supp p ϕ s q Ă Z s .(5) p ϕ ` ¨ ¨ ¨ ` ϕ S qp x q “ , for all x P K .Proof. We can follow the proof of Lemma 3.2 taking K “ L and adapting the steps fromthat proof, keeping in mind that now Z S ` “ H (and hence there is not such a point z S ` ), and that the mapping ϕ S ` is identically 0, and hence, it can be omitted. (cid:3) The following result provides the promised approximation property of C p L q spaces andtheir duals. Theorem 3.4.
Let L be a locally compact space. Given t f , . . . , f (cid:96) u Ă C p L q such that } f j } ď for j “ , . . . , (cid:96) , and given t µ , . . . , µ n u Ă C p L q ˚ with } µ j } ď for j “ , . . . , n ,for each ε ą there exists a norm one projection P : C p L q ÝÑ C p L q satisfying:(1) } P ˚ p µ j q ´ µ j } ă ε , for j “ , . . . , n ,(2) } P p f j q ´ f j } ă ε , for j “ , . . . , (cid:96) ,(3) P p C p L qq is isometrically isomorphic to (cid:96) p for some p P N . Let us comment that this result extends [13, Lemma 3.4] (which, actually, was itself anextension of [4, Proposition 3.2] and [17, Proposition 3.2]). The main difference is thathere we are able to deal with an arbitrary number of functions of C p L q in (2), while inthat lemma only one function is controlled, and besides, this was done with the help ofan inclusion operator which is not the canonical one. However, this difference is crucial inorder to apply Lemma 2.1 (or even its consequence Proposition 2.2).The following observation on the theorem is worth mentioning. Remark 3.5.
Let us observe that by just conveniently ordering the obtained projections inTheorem 3.4, we actually get the following: given a Hausdorff locally compact topologicalspace L , there is a net t P α u α P Λ of norm-one projections on C p L q , converging in the strongoperator topology to the identity operator, such that t P ˚ α u α P Λ converges in the strong N THE BPBP-NU FOR COMPACT OPERATORS 15 operator topology to the identity on C p L q ˚ , and such that P α p C p L qq is isometricallyisomorphic to a finite-dimensional (cid:96) space. Proof of Theorem 3.4.
We will assume first that L is not compact. Since f j P C p L q , j “ , . . . , (cid:96) , there exists a compact set K Ă L such thatsup j “ ,...,(cid:96) t| f j p x q| : x P L z K u ă ε . For each x P K , there exists a relatively compact open subset U x of L containing x andsuch that | f j p x q ´ f j p y q| ă ε y P U x and j “ , . . . , (cid:96) .Therefore, t U x u x P K is a cover of K , and so, there exist a finite subcover t U , . . . , U R ´ u such that K Ď U Y . . . Y U R ´ , and if x, y P U r for some r , then | f j p x q ´ f j p y q| ă ε , for j “ , . . . , (cid:96) .We define µ : “ ř nj “ | µ j | P C p L q ˚ . Since for each j P t , . . . , n u µ j is absolutelycontinuous with respect to µ , by the Radon-Nikod´ym Theorem, there exists g j P L p µ q such that µ j “ g j µ , that is, µ j p f q : “ ż L f d µ j “ ż L f p x q g j p x q d µ p x q for all f P C p L q .Since the set of simple functions is dense in L p µ q , we may choose a set of simple functions t s j : j “ , . . . , n u such that } g j ´ s j } ă ε for j “ , . . . , n .Next, we consider a family t A m u Mm “ of pairwise disjoint measurable sets with µ p A m q ą m , such that each A m is contained in one of the elements of the following coverof L : t U , . . . , U R ´ , L z K u , and also t α m,j : m “ , . . . , M, j “ , . . . , n u such that s j “ ř Mm “ α m,j χ A m . This cover satisfies that if x, y P L z K , or if x, y P U r , then | f j p x q ´ f j p y q| ă ε for all j “ , . . . , (cid:96) and all r “ , . . . , R ´
1. Let C ą max t| α m,j | : m “ , . . . , M, j “ , . . . , n u .Since µ is regular, for each 1 ď m ď M , we can find a compact set K m Ă A m such that µ p A m z K m q ă ε MC and µ p K m q ą m “ , . . . , M .Let K “ K Y K Y . . . Y K M . As K z ˆ R ´ Ť r “ U r ˙ is a compact subset of L , we cancover it with finitely many relatively compact open subsets of L z K that we will denote U R , U R ` , . . . , U P . If we now apply Lemmas 3.1 and 3.2 to the family t U , . . . , U P u andthe compacts t K , . . . , K M u and K , we obtain a refinement of relatively compact opensubsets of L , t Z , . . . , Z S u with K m Ă Z m for m “ , . . . , M and t Z , . . . Z M u pairwisedisjoint, and defining Z S ` to be the set L z ˆ S Ť s “ Z s ˙ , we also have a partition of the unitysubordinated to t Z s u Ss “ , t ϕ s u S ` s “ , such that:(i) t ϕ , . . . , ϕ M u have disjoint support.(ii) ϕ m p K m q ” m “ , . . . , M .(iii) For all M ă s ď S `
1, there exists z s P Z s such that ϕ s p z s q “ s “ , . . . , S `
1, supp p ϕ s q Ă Z s .(v) p ϕ ` . . . ` ϕ S qp K q ” Now, we define P : C p L q ÝÑ C p L q by P p f q : “ M ÿ m “ µ p K m q ˆż K m f d µ ˙ ϕ m ` S ` ÿ s “ M ` f p z s q ϕ s , for all f P C p L q .Let us first check that (2) holds, that is, that } P p f j q ´ f j } ă ε for all j “ , . . . , (cid:96) . Let x P L . We will distinguish two cases: ‚ Case 1: if x P M Ť m “ Z m , then there exists exactly one m such that x P Z m . Then,for each j “ , . . . , (cid:96) , we have: | P p f j qp x q ´ f j p x q| “ ˇˇˇˇˇ P p f j qp x q ´ M ÿ m “ f j p x q ϕ m p x q ´ S ` ÿ s “ M ` f j p x q ϕ s p x q ˇˇˇˇˇ ď ˇˇˇˇˇ µ p K m q ˜ż K m f j p y q d µ p y q ¸ ´ f j p x q ˇˇˇˇˇ ϕ m p x q loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon (I) `` S ` ÿ s “ M ` | f j p x q ´ f j p z s q| ϕ s p x q looooooooooooooooomooooooooooooooooon (II) . For (I), we have(I) “ ˇˇˇˇˇ µ p K m q ˜ż K m p f j p y q ´ f j p x qq d µ p y q ¸ˇˇˇˇˇ ϕ m p x q ď µ p K m q ż K m ε µ p y q “ ε . Now, for (II), let s P t M ` , . . . , S ` u . Note that if x R Z s , then ϕ s p x q “ x P Z s , we have that | f j p x q ´ f j p z s q| ă ε and ř S ` s “ M ` ϕ s p x q ď
1, and so,(II) ă ε . Therefore, | P p f j qp x q ´ f j p x q| ă ε for all x P M Ť m “ Z m , for all j “ , . . . , (cid:96) . ‚ Case 2: if x R M Ť m “ Z m , then for each j “ , . . . , (cid:96) , we have | P p f j qp x q ´ f j p x q| “ ˇˇˇˇˇ S ` ÿ s “ M ` p f j p x q ´ f j p z s qq ϕ s p x q ˇˇˇˇˇ ă ε } P p f j q ´ f j } ă ε for all j “ , . . . , (cid:96) , getting thus (2).Now we check (1), that is, that } P ˚ p µ j q ´ µ j } ă ε for all j “ , . . . , n . Indeed, firstobserve that if ν is a regular Borel (real or complex) measure on L , its associated x ˚ ν P C p L q ˚ is defined as x ˚ ν p f q : “ ż L f p x q d ν p x q , @ f P C p L q , N THE BPBP-NU FOR COMPACT OPERATORS 17 and we identify x ˚ ν ” ν . In our case, we have that P ˚ p ν qp f q “ ż L P p f qp x q d ν p x q“ ż L ˜ M ÿ m “ µ p K m q ˆż K m f d µ ˙ ϕ m p x q ¸ d ν p x q ` ż L ˜ S ` ÿ s “ M ` f p z s q ϕ s p x q ¸ d ν p x q“ M ÿ m “ µ p K m q ˆż K m f d µ ˙ ż L ϕ m p x q d ν p x q ` S ` ÿ s “ M ` f p z s q ż L ϕ s p x q d ν p x q . In particular, if supp p ν q Ă M Ť m “ K m , then by Lemma 3.1.(1) S ` ÿ s “ M ` f p z s q ż L ϕ s p x q d ν p x q ” , @ f P C p L q . Let now ν j : “ t j µ , where t j : “ ř Mm “ α m,j χ K m , for all j “ , . . . , n , that is, ν j p f q “ ż L f p x q ˜ M ÿ m “ α m,j χ K m p x q ¸ d µ p x q , @ f P C p L q . It holds that P ˚ p ν j q “ ν j for j “ , . . . , n . Indeed, as supp p ν j q Ă M Ť m “ K m , we have P ˚ p ν j qp f q “ M ÿ m “ µ p K m q ˆż K m f d µ ˙ ż L ϕ m p x q ˜ M ÿ l “ α l,j χ K l p x q ¸ d µ p x q“ M ÿ m “ µ p K m q ˆż K m f d µ ˙ ż L α m,j χ K m p x q d µ p x q loooooooooooomoooooooooooon α m,j µ p K m q “ ż L f p x q ˜ M ÿ m “ α m,j χ K m p x q ¸ d µ p x q “ ν j p f q for all f P C p L q and all j “ , . . . , n .Now, we know that } P ˚ } “ } P } ď P p ϕ j q “ ϕ j for j “ , . . . , n , we get that } P ˚ } “
1. Therefore, since P ˚ p ν j q “ ν j , we get } P ˚ p µ j q ´ µ j } ď } P ˚ p µ j ´ ν j q} ` } ν j ´ µ j }ď } P ˚ } ¨ } µ j ´ ν j } ` } µ j ´ ν j } ď } µ j ´ ν j } . But we have } µ j ´ ν j } “ } g j µ ´ t j µ } ď } g j µ ´ s j µ } ` } s j µ ´ t j µ }“ } g j ´ s j } ` } s j ´ t j } ă ε ` ε “ ε , since } s j ´ t j } “ ż L ˇˇˇˇˇ M ÿ m “ α m,j χ A m ´ M ÿ m “ α m,j χ K m ˇˇˇˇˇ d µ ď M ÿ m “ | α m,j | loomoon ď C µ p A m z K m q ă M Cε M C “ ε , for all j “ , . . . , n . Hence, } P ˚ p µ j q ´ µ j } ď } µ j ´ ν j } ă ε “ ε for j “ , . . . , n .Let us finish the proof by checking (3). As µ p K m q ą
0, we have K m ‰ H , m “ , . . . , M .Hence, we have that z s P Z s for s “ , . . . , S ` z s R Ť s ‰ s Z s for all s “ , . . . , S `
1. By the definition of P , we have that P p C p L qq “ span t ϕ s : s “ , . . . , S ` u and we will be done by proving the following equality: ›› a ϕ ` ¨ ¨ ¨ ` a S ` ϕ S ` ›› “ max t| a | , . . . , | a S ` |u “ } a } for every a “ p a , . . . , a S ` q . Indeed, for x P L ˇˇ a ϕ p x q ` ¨ ¨ ¨ ` a S ` ϕ S ` p x q ˇˇ ď } a } S ` ÿ s “ ϕ s p x q “ } a } . But for each s , ˇˇ a ϕ p z s q ` ¨ ¨ ¨ ` a S ` ϕ S ` p z s q ˇˇ “ | a s | , and then, ›› a ϕ ` . . . ` a S ` ϕ S ` ›› ě } a } . Hence, the mapping ρ : (cid:96) S ` ÝÑ C p L q given by p a , . . . , a S ` q ÞÝÑ a ϕ ` . . . ` a S ` ϕ S ` is an isometry, and therefore, P p C p L qq is isometrically isomorphic to (cid:96) S ` .Now, for the case when L is compact, by taking K “ L and using Lemma 3.3 insteadof Lemma 3.2, a similar proof is valid, except that now all the elements depending on S ` Z S ` “ H (hence z S ` does not exist), ϕ S ` ” a will only have S components; therefore P p C p L qq is isometrically isomorphic to (cid:96) S in this case. (cid:3) We are now ready to prove the main result of the paper.
Proof of Theorem 1.6.
Let f , . . . , f (cid:96) P B C p L q , µ , . . . , µ n P B p C p L qq ˚ and ε ą P : C p L q ÝÑ C p L q be the projection from Theorem 3.4, which satisfies that P p C p L qq is isometrically isomorphic to (cid:96) p for some p P N . Let r P : X ÝÑ r P p C p L qq bethe operator such that r P p f q “ P p f q for all f P C p L q , and let i : r P p C p L qq ÝÑ C p L q bethe natural inclusion. Let η be the mapping with which all (cid:96) n spaces has the BPBp-nu forcompact operators (see Corollary 2.4). Since n p C p L qq “ n p (cid:96) np q “ n P N (see[21, Proposition 1.11] for instance), in particular, n k p P p C p L qqq “ n k p C p L qq “
1. There-fore, we are in the conditions to apply Lemma 2.1 and get that C p L q has the BPBp-nufor compact operators, as desired. N THE BPBP-NU FOR COMPACT OPERATORS 19
Alternatively, by Remark 3.5, we may prove Theorem 1.6 applying Proposition 2.2instead of Lemma 2.1. (cid:3)
Acknowledgment.
The authors would like to thank Bill Johnson for kindly answeringseveral inquiries.
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Departamento de An´alisis Matem´atico, Universidad de Valencia, DoctorMoliner 50, 46100 Burjasot (Valencia), Spain ORCID:
Email address : [email protected] (Manuel Maestre) Departamento de An´alisis Matem´atico, Universidad de Valencia, DoctorMoliner 50, 46100 Burjasot (Valencia), Spain ORCID:
Email address : [email protected] (Miguel Mart´ın) Departamento de An´alisis Matem´atico, Facultad de Ciencias, Universidadde Granada, 18071 Granada, Spain. ORCID:
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