PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract

We study the Bishop-Phelps-Bollob\'as property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\ell_1$ have the BPBp-nu for compact operators.
ON THE COMPACT OPERATORS CASE OF THE
BISHOP–PHELPS–BOLLOB ´
AS PROPERTY FOR NUMERICAL
RADIUS
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 C0pLqspaces have the BPBp-nu for compact operators for every Haus-
dorff topological locally compact space L. To this end, on the one hand, we provide
some techniques allowing to pass the BPBp-nu for compact operators from subspaces to
the whole space and, on the other hand, we prove some strong approximation property
of C0pLqspaces and their duals. Besides, we also show that real Hilbert spaces and
isometric preduals of `1have the BPBp-nu for compact operators.
1. Introduction, notation, and known results
First we fix some notation in order to be able to describe our aims and results with
precision. Given a Banach space Xover the field Kof real or complex numbers, we
denote by X˚,BX, and SX, its topological dual, its closed unit ball, and its unit sphere,
respectively. If Yis another Banach space, LpX, Y qrepresents the space of all bounded and
linear operators from Xto Y, and we denote by KpX, Y qthe space of compact operators
from Xto Y. When YX, we shall simply write LpXq “ LpX, X qand KpXq “ KpX, Xq.
Given a locally compact Hausdorff topological space L,C0pLqis the Banach space of all
scalar-valued continuous functions on Lvanishing at infinity.
Given an operator TPLpXq, its numerical radius is defined as
νpTq:sup t|x˚pTpxqq|:px, x˚q P ΠpXqu ,
where ΠpXq:“ tpx, x˚q P SXˆSX˚:x˚pxq “ 1u. It is immediate that νpTq ď }T}for
every TPLpXqand that νis a seminorm on LpXq. Very often, νis actually a norm
on LpXqequivalent to the usual operator norm. The numerical index of the space X
measures this fact and it is given by
npXq:inftνpTq:TPLpXq,}T} “ 1u
maxtkě0: k}T} ď νpTq,@TPLpXqu.
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; numerical
radius 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 de
Analuc´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 by
MINECO and FEDER project MTM2017-83262-C2-1-P.
1
arXiv:2102.10937v1 [math.FA] 22 Feb 2021
2 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
It is clear that 0 ďnpXq ď 1 and that npXq ą 0 if and only if the numerical radius is a
norm on LpXqequivalent to the operator norm. As in this paper we will mainly deal with
compact operators, we will also need the following concept from [11]. Given a Banach
space X, the compact numerical index of Xis
nKpXq:inftνpTq:TPKpXq,}T} “ 1u
maxtkě0: k}T} ď νpTq,@TPKpXqu.
We refer the reader to [11], [18], [19], [21, Subsection 1.1], and references therein for more
information and background.
An operator SPLpX, Y qis said to attain its norm whenever there exists some xPSX
such that }S}“}Spxq}. An operator TPLpXqis said to attain its numerical radius
whenever there exists some px, x˚q P ΠpXqsuch that νpTq“|x˚pTpxqq|. The sets of norm
attaining operators from Xto Yand of numerical radius attaining operators on Xwill
be denoted, respectively, by NApX, Y qand NRApXq.
In 1961, Bishop and Phelps [8] proved that the set NApX, Kqof norm attaining func-
tionals on a Banach space Xis always dense in X˚. However, this result has been shown
to 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 density
of 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 that
you can approximate simultaneously a functional and a point where it almost attains
its norm by a norm-attaining funcional and a point where the new functional attains
its norm, respectively. In order to extend Bollob´as’ result to norm attaining operators
between Banach spaces, Acosta, Aron, Garc´ıa and Maestre [3] introduced in 2008 the
Bishop-Phelps-Bollob´as property as follows.
Definition 1.1 ([3]).A pair of Banach spaces pX, Y qhas the Bishop-Phelps-Bollob´as
property (BPBp for short) if given εP p0,1q, there exists ηpεqPp0,1qsuch that whenever
TPLpX, Y qand x0PSXsatisfy }T} “ 1 and }Tpx0q} ą 1´ηpεq, there are SPLpX, Y q
and x1PSXsuch that
}S}“}Spx0q} “ 1,}x0´x1} ă ε, }S´T} ă ε.
If the above property holds when we restrict the operators Tand Sto be compact, we
say that the pair pX, Y qhas 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 pX, Kqhas the
Bishop-Phelps-Bollob´as property for every Banach space X. In the paper [3] a variety of
pairs of spaces satisfying the BPBp are provided, together with examples of pairs pX, Y q
of Banach spaces failing the BPBp for which NApX, Y qis dense in LpX, Y q. We refer the
reader to the survey [2] and the paper [6] for more information and background on the
BPBp.
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 Xis said to have the Bishop-Phelps-Bollob´as
property for numerical radius (BPBp-nu for short) if for every 0 ăεă1, there exists
ηpεq P p0,1qsuch that whenever TPLpXqand px, x˚q P ΠpXqsatisfy νpTq “ 1 and
ON THE BPBP-NU FOR COMPACT OPERATORS 3
|x˚pTpxqq| ą 1´ηpεq, there exist SPLpXqand py, y˚q P ΠpXqsuch that
νpSq“|y˚pSpyqq| “ 1,}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 that
property. We summarize next some of the most important results on the matter:
(1) The spaces c0and `1have the BPBp-nu [16].
(2) L1pRqhas the BPBp-nu [14].
(3) Finite-dimensional spaces have the BPBp-nu [20].
(4) The real or complex space Lppµqhas the BPBp-nu for every measure µwhen
1ďpă 8 ([20, Example 8] except for the real case p2, which is covered in [21,
Corollary 3.3]).
(5) Any uniformly convex and uniformly smooth Banach space Xwith npXq ą 0 has
the BPBp-nu [20].
(6) Every separable infinite-dimensional Banach space can be renormed to fail the
BPBp-nu [20], even though the set of numerical radius attaining operators is always
dense in spaces with the Radon-Nikod´ym property.
(7) The real space CpKqhas the BPBp-nu under some extra conditions on the compact
Hausdorff space K(for example, when Kis metrizable) [7]. Let us comment that
it remains unknown if the result is true for all compact Hausdorff spaces or what
happens 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 attaining
compact operators by Johnson and Wolfe [17]) which allow to carry the BPBp for com-
pact operators from sequence spaces (such as c0and `p) to function spaces (as C0pLqand
Lppµq). As one of the main results, it is shown in [13] that the BPBp for compact opera-
tors of the pair pc0, Y qis sufficient to get the BPBp for compact operators of all the pairs
pC0pLq, Y qregardless of the locally compact Hausdorff topological space L.
Our aim in this paper is to study the following property, inspired both by the BPBp
for compact operators and by the BPBp for numerical radius.
Definition 1.3. A Banach space Xis said to have the BPBp-nu for compact operators
if for every 0 ăεă1, there exists ηpεqPp0,1qsuch that whenever TPKpXqand
px, x˚q P ΠpXqsatisfy νpTq “ 1 and |x˚pTpxqq| ą 1´ηpεq, there exist SPKpXqand
py, y˚q P ΠpXqsuch that
νpSq“|y˚pSpyqq| “ 1,}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 the
BPBp-nu for subspaces of LpXqwas defined and studied in the case of L1pµq, with µa
finite measure. Let us provide a list of spaces that are known to have the BPBp-nu for
compact operators.
Examples 1.4. The following spaces have the BPBp-nu for compact operators:
(a) Finite dimensional spaces [20, Proposition 2].
(b) c0and `1(adapting the proofs given in [16, Corollaries 3.3 and 4.2]).
4 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
(c) L1pµqfor 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 Lppµqspaces have the
BPBp-nu for compact operators when 1 ăpă 8. However, the adaptation to the
compact 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. Lppµqhas the BPBp-nu for compact operators, for every measure µ
and 1ăpă 8.
Our main objective in this paper is to prove the following result, which is not covered
by Examples 1.4.
Theorem 1.6. If Lis a locally compact Hausdorff space, then C0pLqhas the BPBp-nu
for compact operators.
As a consequence, we get that L8pµqspaces have the BPBp-nu for compact operators,
completing Example 1.4.c and Proposition 1.5.
Corollary 1.7. L8pµqhas the BPBp-nu for compact operators for every measure µ.
Let us recall that it is shown in [7] that the real space CpKqhas the BPBp-nu for some
compact Hausdorff spaces K, but the general case, as well as the complex case, remain
open as far as we know. However, Theorem 1.6 gives a complete answer in the case of
compact operators.
To get the proof of Theorem 1.6, we need two kind of ingredients. On the one hand,
we provide in Section 2some abstract results that will allow us to carry the BPBp-nu for
compact operators from sequence spaces into function spaces, in some cases. The more
general result of this kind is Lemma 2.1, which will be the first ingredient for the proof of
Theorem 1.6. It is somehow an extension of [13, Lemma 2.1] but it needs more restrictive
hypothesis in order to deal with the numerical radius instead of with the norm of the
operators. We also provide in that section some useful particular cases of Lemma 2.1
which allow to show, for instance, that every predual of `1has the BPBp-nu for compact
operators, see Corollary 2.6. The second ingredient for the proof of Theorem 1.6 is some
strong approximation property of C0pLqspaces and their duals which will be provided in
Section 3(see Theorem 3.4) and which will allow us to apply Lemma 2.1 in this case, thus
providing the proof of Theorem 1.6. Let us also comment that Theorem 3.4 gives a much
stronger approximation property of C0pLqand its dual space than [13, Lemma 3.4].
1.1. Lpspaces have the BPBp-nu for compact operators.
In this subsection, we will adapt the concepts and results from [20] and [21] to the
compact operators case to show that Lppµqspaces have the BPBp-nu for compact operators
for 1 ăpă 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 Xis said to have the weak BPBp-nu for compact oper-
ators if for every 0 ăεă1, there exists ηpεqPp0,1qsuch that whenever TPKpXqand
ON THE BPBP-NU FOR COMPACT OPERATORS 5
px, x˚q P ΠpXqsatisfy νpTq “ 1 and |x˚pTpxqq| ą 1´ηpεq, there exist SPKpXqand
py, y˚q P ΠpXqsuch that
νpSq“|y˚pSpyqq|,}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 without
asking for the condition νpSq “ 1 (see Definition 1.3).
It is shown in [20, Proposition 4] that uniformly convex and uniformly smooth Banach
spaces have the weak BPBp-nu. This result also holds for the compact operators version
by an easy adaptation of the proof.
Proposition 1.9. If a Banach space is uniformly convex and uniformly smooth, then it
has the weak BPBp-nu for compact operators.
Proof. We can follow the proof of [20, Proposition 4], just keeping in mind that if the
original operator T0is compact, then the rest of operators Tnfrom that proof are also
compact, and so, Sis compact too.
Later, in [20, Proposition 6], it is proven that in Banach spaces with positive numerical
index, the BPBp-nu and the weak BPBp-nu are equivalent. This property is also true for
the compact operators versions of the properties if we use the compact numerical index.
Proposition 1.10. Let Xbe a Banach space such that nKpXq ą 0. Then Xhas the
BPBp-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 Tand Sbeing
now compact operators, and using nKpXqinstead of npXq.
As a consequence of these two results, similarly to what is done in [20], we get that all
Lppµqspaces have the BPBp-nu for compact operators when 1 ăpă 8 in the complex
case and when 1 ăpă 8,p2 in the real case. This is so because, on the one hand, in
the real case,
nKpLppµqq ě npLppµqq ą 0`1ăpă 8, p 2˘
by [24] and, on the other hand, nKpXq ě 1{eą0 for every complex Banach space (see
[18, Eq. (1) in p. 156], for instance).
This provides the proof of Proposition 1.5 for all values of pin p1,`8q in the complex
case and for all values of pin p1,`8q except for p2 in the real case.
Our next aim is to show that real Hilbert spaces also have the BPBp-nu for compact
operators, by adapting the ideas from [21].
First, given a real Banach space X, we consider the following subset of KpXq:
ZKpXq: TPKpXq:νpTq “ 0(
which is the set of all skew-hermitian compact operators on X. Observe that
ZKpXq “ KpXq X ZpXq,
where ZpXqis the Lie-algebra of all skew-hermitian operators on X(see [21, p. 1004] for
instance). Adapting the concept of second numerical index given in [21], we define the
6 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
second numerical index for compact operators of a Banach space Xas the constant
n1
KpXq:inf νpTq:TPKpXq,}T`ZKpXq} “ 1(
max Mě0: M}T`ZKpXq} ď νpTqfor all TPKpXq(,
where }T`ZKpXq} is the quotient norm in KpXq{ZKpXq.
The next result is a version for compact operators of [21, Theorem 3.2].
Proposition 1.11. Let Xbe a real Banach space with n1
KpXq ą 0. Then, the BPBp-nu
for 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 of
compact operators. That is: all the involved operators T,S,S1and S2are now compact,
the ZpXqset is replaced by ZKpXq, and the index n1pXqis replaced by n1
KpXq.
We are going to see next that the second numerical index for compact operators of a
real Hilbert space equals one.
Proposition 1.12. Let Hbe a real Hilbert space. Then, n1
KpHq “ 1.
The proof of this result will be an adaptation of the one of [21, Theorem 2.3]. Recall
that in a real Hilbert space endowed with an inner product p¨|¨q,H˚identifies with Hby
the isometric isomorphism xÞÝÑ p¨|xq. Therefore, ΠpXq “ tpx, xq P HˆH:xPSHu,
and so, for every TPLpHq, one has νpTq “ supt|pT x|xq|:xPSHu. We first need to give
the compact operators version of [21, Lemma 2.4] whose proof is an obvious adaptation
of the proof of that result.
Lemma 1.13. Let Hbe a real Hilbert space.
(a) ZKpHq“tTPKpHq:T“ ´T˚u.
(b) If TPKpHqis selfadjoint (i.e. TT˚), then }T} “ νpTq.
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 Tand Sare now compact, and the
set ZpXqis replaced by ZKpXq.
As a consequence of Propositions 1.9,1.11, and 1.12, we get the following result which
provides the proof of the pending part of Proposition 1.5.
Corollary 1.14. If His a real Hilbert space, then it has the BPBp-nu for compact oper-
ators.
2. First ingredient: the tools
In this section, we will provide an abstract result that will allow us later to carry
the BPBp-nu for compact operators from some sequence spaces to function spaces. The
most 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 absolute
norm | ¨ |ais a norm in R2such that |p1,0q|a“ |p0,1q|a1 and |ps, tq|a“ |p|s|,|t|q|afor
every ps, tq P R2. Given a Banach space X, we say that a projection Pon Xis an absolute
projection if there is an absolute norm |¨|asuch that }x} “ ˇˇp}Ppxq},}x´Ppxq}qˇˇafor
ON THE BPBP-NU FOR COMPACT OPERATORS 7
every xPX. Examples of absolute projections are the M- and L-projections and, more
in general, the `p-projections. We refer the reader to [13] for the use of absolute norms
with the Bishop-Phelps-Bollob´as type properties and to the references therein for more
information on absolute norms.
Lemma 2.1. Let Xbe a Banach space satisfying that nKpXq ą 0. Suppose that there is
a mapping η:p0,1q ÝÑ p0,1qsuch that given δą0,x˚
1, . . . , x˚
nPBX˚and x1, . . . , x`P
BX, we can find norm one operators r
P:XÝÑ r
PpXq,i:r
PpXq ÝÑ Xsuch that for
P:i˝r
P:XÝÑ X, the following conditions are satisfied:
(i) }P˚px˚
jq ´ x˚
j} ă δ, for j1, . . . , n.
(ii) }Ppxjq ´ xj} ă δ, for j1, . . . , `.
(iii) r
P˝iId r
PpXq.
(iv) r
PpXqsatisfies the Bishop-Phelps-Bollob´as property for numerical radius for com-
pact operators with the mapping η.
(v) Either Pis an absolute projection and iis the natural inclusion, or nKpr
PpXqq “
nKpXq “ 1.
Then, Xsatisfies 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 with
one point. Second, the requirements of item (v) on the compact numerical index or on the
absoluteness of the projections did not appear in [13, Lemma 2.1], but they are needed
here as numerical radius does not behave well in general with respect to extensions of
operators.
Proof. Given εP p0,1q, let ε0pεqbe the unique number with 0 ăε0pεq ă 1 such that
ε0pεqˆ2
3`1
p1´ε0pεqq nKpXq˙ε,
which, in particular, satisfies that ε0pεq ă ε. From now on, we shall simply write ε0instead
of ε0pεq. We define next
(1) η1pεq:min #ε2
0pnKpXqq2
72 ,`η`ε0
3˘˘2pnKpXqq2
72 +`εP p0,1q˘,
where ηis the function appearing in the hypotheses of the lemma. We fix TPKpXqwith
νpTq “ 1 (thus, }T} ď 1
nKpXq) and px1, x˚
1q P ΠpXqsuch that
|x˚
1pTpx1qq| ą 1´η1pεq.
Since T˚pBX˚qis relatively compact, we can find x˚
2, . . . , x˚
nPBX˚such that
min
2ďjďn}T˚px˚q ´ x˚
j} ă η1pεqfor all x˚PBX˚.
Similarly, since TpBXqis relatively compact, we can find x2, . . . , x`PBXsuch that
min
2ďjď`}Tpxq ´ xj} ă η1pεqfor all xPBX.
Let r
P:XÝÑ r
PpXq,i:r
PpXq ÝÑ Xand P:i˝r
P:XÝÑ Xsatisfying the conditions
(i)-(v) for x1, . . . , x`PBX,x˚
1, . . . , x˚
nPBX˚and δη1pεq.
8 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
Now, for every x˚PBX˚, we have
}T˚px˚q ´ P˚pT˚px˚qq}
ďmin
2ďjďn }T˚px˚q ´ x˚
j}`}x˚
j´P˚px˚
jq} ` }P˚px˚
jq ´ P˚pT˚px˚qq}(ă3η1pεq,
and hence, }T´T P } “ }T˚´P˚T˚} ď 3η1pεq. On the other hand, for each xPBX, we
have
}Tpxq ´ PpTpxqq} ď min
2ďjď`t}Tpxq ´ xj}`}xj´Ppxjq} ` }Ppxjq ´ PpTpxqq}u
ă3η1pεq,
and then, }T´P T } ď 3η1pεq. Therefore,
}PTP ´T}ď}PTP ´P T }`}P T ´T} ď }T P ´T}`}P T ´T} ď 6η1pεq.
Consider pr
Ppx1q, i˚px˚
1qq P r
PpXq ˆ p r
PpXqq˚. Note that it is not true in general that
pr
Ppx1q, i˚px˚
1qq P Πpr
PpXqq, but we have that }r
Ppx1q} ď 1, }i˚px˚
1q} ď 1, and also, that
x˚
1pipr
Ppx1qqq “ x˚
1px1q
loomoon
1
´x˚
1pipr
Ppx1qq ´ x1q
loooooooooomoooooooooon
}P x1´x1η1pεq
ùñ Repx˚
1pipr
Ppx1qqqq ě 1´η1pεq.
By the Bishop-Phelps-Bollob´as Theorem (see [10, Corollary 2.4.b] for this version), there
exist py, y˚q P Πpr
PpXqq satisfying that
max !}y´r
Ppx1q},}y˚´i˚px˚
1q})ďa2η1pεq ď ε0
3.
Next, we observe that the following two inequalities hold:
}r
P˚py˚q ´ x˚
1} ď } r
P˚py˚q ´ r
P˚pi˚px˚
1qq} ` } r
P˚pi˚px˚
1qq ´ x˚
1}
ďa2η1pεq ` η1pεq ď 2
3ε0.
(2)
(3) }ipyq ´ x1} ď }ipyq ´ ipr
Ppx1qq} ` }ipr
Ppx1qq ´ x1} ď a2η1pεq ` η1pεq ď 2
3ε0.
Let T1:r
P˝T˝i:r
PpXq ÝÑ r
PpXq.
Claim. We have that
|y˚pT1yq| ą 1´η´ε0
3¯and |y˚pT1yq| ą 1´ε0.
Indeed, from equations (2) and (3), we obtain that
ˇˇx˚pTpx1qq´ r
P˚py˚pTpipyqqqqˇˇ
ď |x˚
1pTpx1qq ´ x˚
1pTpipyqqq| ` |x˚
1pTpipyqqq ´ r
P˚py˚pTpipyqqqq|
ď }T}}x1´ipyq} ` }T}}x˚
1´r
P˚py˚q}
ď2}T}´a2η1pεq ` η1pεq¯.
Now, we can estimate |y˚pT1pyqq| as follows:
|y˚pT1pyqq| “ ˇˇr
P˚py˚pTpipyqqqqˇˇ
졡x˚
1pTpx1qqˇˇ´ˇˇx˚
1pTpx1qq ´ r
P˚py˚pTpipyqqqqˇˇ
ě1´η1pεq ´ 2}T}a2η1pεq ´ 2}T}η1pεq.
ON THE BPBP-NU FOR COMPACT OPERATORS 9
From here, using the definition of η1pεqgiven in Eq. (1) and the fact that }T} ď 1{nKpXq,
we get both assertions of the claim.
In particular, we get that νpT1q ě 1´ε0ą0. On the other hand, we also have that
νpT1q ď 1. Indeed, if there were some pq, q˚q P Πpr
PpXqq with |q˚pT1pqqq| ą 1, we would
get
|q˚pT1pqqq| “ |q˚pr
PpTpipqqqqq| “ |p r
P˚pq˚qqpTpipqqqq| ą 1,
but νpTq “ 1, and
pr
P˚pq˚qqpipqqq “ q˚pr
Ppipqqqq “ q˚pqq “ 1.
Thus pipqq,r
P˚pq˚qq P ΠpXq, and that is a contradiction.
We define now the operator r
T:T1
νpT1q. Clearly, r
Tis a compact operator such that
νpr
Tq “ 1. From the claim, we get that
ˇˇy˚pr
Tpyqqˇˇ1
νpT1q|y˚pT1pyqq| ě |y˚pT1pyqq| ą 1´η´ε0
3¯.
Now, since r
PpXqhas the BPBp-nu for compact operators with the mapping η, there exist
a compact operator r
S:r
PpXq ÝÑ r
PpXqwith νpr
Sq “ 1 and pz, z˚q P Πpr
PpXqq such that
νpr
Sq “ ˇˇz˚pr
Spzqqˇˇ1,}z´y} ă ε0
3,}z˚´y˚} ă ε0
3,}r
S´r
T} ă ε0
3.
Let tipzq P BXand t˚r
P˚pz˚q P BX˚. We have that
t˚ptq “ z˚pr
Ppipzqqq “ z˚pzq “ 1.
Thus pt, t˚q P ΠpXq, and also, by (2) and (3),
}t´x1} ď }t´ipyq} ` }ipyq ´ x1}“}ipzq ´ ipyq} ` }ipyq ´ x1} ă ε0
3`2ε0
3ε0ďε,
}t˚´x˚
1} ď } r
P˚pz˚q ´ r
P˚py˚q} ` } r
P˚py˚q ´ x˚
1} ă ε0
3`2ε0
3ε0ďε.
We define Si˝r
S˝r
P:XÝÑ X, which is a compact operator. It is clear that νpSq ě 1
since
|t˚pSptqq| “ |z˚pr
Ppipr
Spr
Ppipzqqqqqq| “ |z˚pr
Spzqq| “ 1.
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´PTP}`}PTP ´T}
“ }i˝r
S˝r
P´i˝r
T˝r
P} `
PTP
νpT1q´PTP` }PTP ´T}
ď } r
S´r
T}`}T} ¨ ˇˇˇˇ
1
νpT1q´1ˇˇˇˇ` }PTP ´T}
and, since }T} ď 1
nKpXq, 1 ´ε0ďνpT1q ď 1, and 6η1pεq ď ε0
3, we continue as:
ďε0
3`ε0
p1´ε0qnKpXq`6η1pεq ď ε0ˆ2
3`1
p1´ε0qnKpXq˙ăε.
We finish the proof if we prove that νpSq ď 1. We consider the following cases:
10 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
Case 1: if r
Pis an absolute projection and iis the natural inclusion, as a conse-
quence of [11, Lemma 3.3], we get that
νpSq “ νpi˝r
S˝r
Pq “ νpr
Sq “ 1.
Case 2: if nKpXq “ nKpr
PpXqq “ 1, then
νpSq“}S}ď}r
S} “ νpr
Sq “ 1.
Hence, the result follows in the two cases.
We will now provide some applications and consequences of the previous lemma. Given
a continuous projection P:XÝÑ X, if we set r
P:XÝÑ r
PpXq “ PpXq Ă X(that is,
r
Pis just the operator Pwith a restricted codomain) and i:PpXq ÝÑ Xis the natural
inclusion then, trivially, we have that Pi˝r
Pand that r
P˝iId r
PpXq. This easy
observation allows to get the following particular case of Lemma 2.1.
Proposition 2.2. Let Xbe a Banach space with nKpXq ą 0. Suppose that there exists a
net tPαuαPΛof norm-one projections on Xsatisfying that tPαpxqu ÝÑ xfor all xPXand
tP˚
αpx˚qu ÝÑ x˚for all x˚PX˚, and that there exists a function η:p0,1q ÝÑ p0,1qsuch
that all the spaces PαpXqwith αPΛhave the BPBp-nu for compact operators with the
function η. Suppose, moreover, that for each αPΛ, at least one of the following conditions
is satisfied:
(1) the projection Pαis absolute,
(2) nKpPαpXqq “ nKpXq “ 1.
Then, the space Xhas the BPBp-nu for compact operators.
We may now obtain the following consequence of the above result. Given a Banach
space Xand mPN, the space `m
8pXqrepresents the `8-sum of mcopies of X, and we will
write `8pXqfor the `8-sum of countably infinitely many copies of X. Similarly, c0pXqis
the c0-sum of countably infinitely many copies of X. When XK, we just write `m
8for
`m
8pKq.
Corollary 2.3. Let Xbe a Banach space with nKpXq ą 0. Then, the following statements
are equivalent:
(i) The space c0pXqhas the BPBp-nu for compact operators.
(ii) There is a function η:p0,1q ÝÑ p0,1qsuch that all the spaces `n
8pXq, with nPN,
have the BPBp-nu for compact operators with the function η.
Moreover, if Xis finite dimensional, these properties hold whenever c0pXqor `8pXqhave
the BPBp-nu.
Proof. That (ii) implies (i) is a consequence of Proposition 2.2 since for every nPN, the
operator on c0pXqwhich is the identity on the first ncoordinates and 0 elsewhere is an
absolute projection whose image is isometrically isomorphic to `n
8pXq.
(i) implies (ii) is a consequence of [12, Proposition 4.3], as one can easily see `n
8pXq
as an `8-summand of c0pXq. Let us comment that the function ηvalid for all `n
8pXqis
the function valid for c0pXqand this actually follows from the proof of [12, Theorem 4.1]
(from which [12, Proposition 4.3] actually follows).
ON THE BPBP-NU FOR COMPACT OPERATORS 11
Finally, when Xhas finite dimension, if c0pXqor `8pXqhas the BPBp-nu, then con-
dition (ii) holds by using [12, Theorem 4.1] and the fact that `n
8pXqis finite-dimensional
and so, every operator from `n
8pXqto itself is compact.
As stated in Examples 1.4, that c0and the spaces `n
8for nPNhave the BPBp-nu for
compact operators is a consequence of [16, Corollary 4.2] and [20, Proposition 2]. Actually,
the fact that all the space `n
8have the BPBp-nu with the same function ηfollows from
[16, Corollary 4.2] and (the proof of) [12, Theorem 4.1]. However, let us note that we can
also get this result as a consequence of our previous corollary.
Corollary 2.4. There is a function η:p0,1q ÝÑ p0,1qsuch that the space c0and the
spaces `n
8with nPN, have the BPBp-nu for compact operators with the function η.
Additionally, [12, Proposition 4.3] also implies that whenever `n
8pXqhas the BPBp-nu
for compact operators for some nPN, then so does X, although the converse remains
unknown in general (even for n2).
Another consequence of Proposition 2.2 is the following:
Corollary 2.5. Let Xbe a Banach space with nKpXq ą 0. Suppose that there exists a
net tPαuαPΛof norm-one projections on Xsuch that αĺβimplies PαpXq Ă PβpXq, that
tP˚
αpx˚qu ÝÑ x˚for all x˚PX˚, and that there exists a function η:p0,1q ÝÑ p0,1qsuch
that all the spaces PαpXqwith αPΛhave the BPBp-nu for compact operators with the
function η. Suppose, moreover, that for each αPΛ, at least one of the following conditions
is satisfied:
(1) the projection Pαis absolute,
(2) nKpPαpXqq “ nKpXq “ 1.
Then, the space Xhas the BPBp-nu for compact operators.
Proof. Observe that in order to apply Proposition 2.2 we only need that tPαxu ÝÑ xin
norm for all xPX. But this is proved in [13, Corollary 2.4], so we are done.
The previous result can be used to prove that all the preduals of `1have the BPBp-nu
for compact operators.
Corollary 2.6. Let Xbe a Banach space such that X˚is isometrically isomorphic to `1.
Then Xhas the BPBp-nu for compact operators.
Proof. By using a deep result due to Gasparis [15], it is shown in the proof of [13, Theorem
3.6] that there exists a sequence of norm-one projections Pn:XÝÑ Xsatisfiying that
Pn`1PnPn(and so, PnpXq Ă Pn`1pXq), that PnpXqis isometrically isomorphic to `n
8,
and also that P˚
npx˚q ÝÑ x˚for all x˚PX˚(this claim holds since the sets Yndefined on
that proof satisfy that their union is dense in X˚`1).
Next, as PnpXqis isometrically isomorphic to `n
8, on the one hand we have that all
the spaces PnpXqhave the BPBp-nu for compact operators with the same function ηas
a consequence of Corollary 2.4. On the other hand, npXq “ npPnpXqq “ 1 for all nPN
(see [18], for instance) so, in particular, nKpXq “ nKpPnpXqq “ 1 for all nPN. Finally,
Corollary 2.5 provide the desired result.
12 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
3. Second ingredient: a strong approximation property of C0pLqspaces and
their duals
The aim of this section is to provide some strong approximation property of C0pLq
spaces and their duals which allow to use Lemma 2.1 (actually, Proposition 2.2) to give a
proof of Theorem 1.6. We need a number of technical lemmas.
Lemma 3.1. Let Lbe a locally compact space, let tK1, . . . , KMube a family of pairwise
disjoint non-empty compact subsets of L, and let KĂLbe a compact set with
M
Ť
m1
KmĂ
K. If tU1, . . . , URuis a family of relatively compact open subsets of Lcovering Ksuch that
for each mthere is an rpmqwith KmĂUrpmq,m1, . . . , M , then there exists an open
refinement tZ1, . . . , ZSu,MďSďR`Mwith Z1, . . . , ZMpairwise disjoint, satisfying:
(1) For m1, . . . , M,KmĂZm, and KmXZs“ H for all sP t1, . . . , Suztmu.
(2) For all s0ąM, there exists zs0PZs0zˆŤ
ss0
Zs˙.
Proof. As tK1, . . . , KMuare pairwise disjoint, there exist tV1,...VMupairwise disjoint
open subsets of Lwith KmĂVmĂUrpmq,m1, . . . , M .
The family "V1, . . . , VM, U1zˆM
Ť
m1
Km˙, . . . , URzˆM
Ť
m1
Km˙*is another cover of K
by open subsets of Lsubordinated to tUruR
r1. We define Zm:Vmfor m1, . . . , M,
and Wr:UrzˆM
Ť
m1
Km˙for r1, . . . , R.
If W1ĂV1Y. . . YVM, then tV1, . . . , VM, W2, . . . , WRuis again a cover of K. If that
happens again and again until WR, we have that tZ1, . . . , ZMuis the cover we were looking
for. In other case, let r1ě1 be the first natural number such that there exists wr1P
Wr1zˆM
Ť
m1
Vm˙, and denote ZM`1:Wr1. The family tV1, . . . , VM, Wr1, Wr1`1, . . . , WRu
is a cover of Kby open sets, and then, so is the family
V1, . . . , VM, Wr1, Wr1`1ztwr1u, . . . , WRztwr1u(.
Consider now r2ąr1the first natural number such that there exists wr2PWr2ztwr1uand
wr2RV1Y. . . YVMYWr1. Let ZM`2:Wr2ztwr1uand proceed as before. In at most R
steps, we get tZ1, . . . , ZSu,MďSďR`M, such that
KmĂZmfor m1,...M.
ˆM
Ť
m1
Km˙XZs“ H for sąM.
For all s0ąM, there exists wrs0´MPZs0zˆŤ
ss0
Zs˙.
We next provide a result showing the existence of certain partitions of the unity. We
separate the non-compact case (Lemma 3.2) and the compact case (Lemma 3.3) for the
sake of clarity. We start with the non-compact case.
Lemma 3.2. Let Lbe a non-compact locally compact space. Let KĂLbe a compact set
and tK1, . . . , KMua family of pairwise disjoint non-empty compact subsets of K. Given a
ON THE BPBP-NU FOR COMPACT OPERATORS 13
family tU1, . . . , URuof relatively compact open subsets of Lthat cover K, let tZ1, . . . , ZSu
be a family of open subsets of Lcovering Ksuch that they satisfy the thesis of Lemma 3.1,
and denote by ZS`1the set LzˆS
Ť
s1
Zs˙. Then, there exists a partition of the unity
subordinated to tZsuS`1
s1,tϕsuS`1
s1, such that:
(1) tϕ1, . . . , ϕMuhave disjoint support.
(2) ϕmpKmq ” 1, for m1, . . . , M .
(3) For all MăsďS`1, there exists zsPZssuch that ϕspzsq “ 1.
(4) For s1, . . . , S `1,supppϕsq Ă Zs.
(5) pϕ1` ¨ ¨ ¨ ` ϕSqpxq “ 1, for all xPK.
Proof. By hypothesis, there exists some zS`1PLzˆS
Ť
s1
Zs˙, since for all s,ZsĂ
R
Ť
r1
Ur,
which is a compact set. Now, we follow the argument from the proof of [25, Theorem
2.13], but adapted to our case.
As KĂZ1Y. . . YZS, for each xPK, there exists a neighbourhood of x,Yx, with
compact closure YxĂZsfor some s. Consider x1, . . . , xpsuch that KĂYx1Y. . . YYxp.
For each 1 ďsďS, let Hsbe the union of those Yxjwhich lie in Zs, and if Măs0ďS,
we take Hs0Y tzs0u, with zs0PZs0zˆŤ
ss0
Zs˙. Note that the sets H1, . . . , HMand
HM`1YtzM`1u, . . . , HSYtzSuare non-empty. By Urysohn’s Lemma, there are continuous
functions gs:LÝÑ r0,1ssuch that gspHsq ” 1 and gsˇˇLzZs0, for 1 ďsďM, and
gs0pHs0Y tz0uq ” 1 and gs0ˇˇLzZs0
0 for Măs0ďS. Define
ϕ1:g1,
ϕ2:“ p1´g1qg2,
.
.
.
ϕS:“ p1´g1qp1´g2q¨¨¨p1´gS´1qgS
Clearly, supppϕsq Ă Zsfor all s1, . . . , S, and we have that
ϕ1` ¨ ¨ ¨ ` ϕS1´ p1´g1q¨¨¨p1´gSq.
Since KĂH1Y. . . YHS, for each xPK, there exists sspxqwith gspxq “ 1, and also,
for all s1, . . . , M , we have that
txPL:ϕspxq ‰ 0uĂtxPL:gspxq ‰ 0u Ă Zs.
Therefore, the functions tϕ1, . . . , ϕMuhave disjoint support, and ϕ1` ¨ ¨ ¨ ` ϕS1 on K.
We define ϕS`1:1´ pϕ1` ¨ ¨ ¨ ` ϕSq “ p1´g1q¨¨¨p1´gSq. Moreover, KmĂZmfor
m1, . . . , M , and KmXZs“ H for ms,m1, . . . , M ,s1, . . . , S. Hence,
ϕmpxq “
S
ÿ
s1
ϕspxq “ 1,@xPKm, m 1, . . . , M.
14 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
On the other hand, if Măs0ďS, let zs0PZs0zˆŤ
ss0
Zs˙. We have that
ϕs0pzs0q “
S
ÿ
s1
ϕspzs0q “ 1,
and
zS`1R
S
ď
s1
Zs,thus ϕS`1pzS`1q “ 1.
The next result is the version of the previous lemma for compact topological spaces.
Lemma 3.3. Let Lbe a compact space. Let tK1, . . . , KMube a family of pairwise disjoint
non-empty compact subsets of L. Given a family tU1, . . . , URuof relatively compact open
subsets of Lthat cover it, let tZ1, . . . , ZSube a family of open subsets of Lcovering K
such that they satisfy the thesis of Lemma 3.1. Then, there exists a partition of the unity
subordinated to tZsuS
s1,tϕsuS
s1, such that:
(1) tϕ1, . . . , ϕMuhave disjoint support.
(2) ϕmpKmq ” 1, for m1, . . . , M .
(3) For all MăsďS, there exists zsPZssuch that ϕspzsq “ 1.
(4) For s1, . . . , S,supppϕsq Ă Zs.
(5) pϕ1` ¨ ¨ ¨ ` ϕSqpxq “ 1, for all xPK.
Proof. We can follow the proof of Lemma 3.2 taking KLand adapting the steps from
that proof, keeping in mind that now ZS`1“ H (and hence there is not such a point
zS`1), and that the mapping ϕS`1is identically 0, and hence, it can be omitted.
The following result provides the promised approximation property of C0pLqspaces and
their duals.
Theorem 3.4. Let Lbe a locally compact space. Given tf1, . . . , f`u Ă C0pLqsuch that
}fj} ď 1for j1, . . . , `, and given tµ1, . . . , µnu Ă C0pLq˚with }µj} ď 1for j1, . . . , n,
for each εą0there exists a norm one projection P:C0pLq ÝÑ C0pLqsatisfying:
(1) }P˚pµjq ´ µj} ă ε, for j1, . . . , n,
(2) }Ppfjq ´ fj} ă ε, for j1, . . . , `,
(3) PpC0pLqq is isometrically isomorphic to `p
8for some pPN.
Let us comment that this result extends [13, Lemma 3.4] (which, actually, was itself an
extension of [4, Proposition 3.2] and [17, Proposition 3.2]). The main difference is that
here we are able to deal with an arbitrary number of functions of C0pLqin (2), while in
that lemma only one function is controlled, and besides, this was done with the help of
an inclusion operator which is not the canonical one. However, this difference is crucial in
order 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 in
Theorem 3.4, we actually get the following: given a Hausdorff locally compact topological
space L, there is a net tPαuαPΛof norm-one projections on C0pLq, converging in the strong
operator topology to the identity operator, such that tP˚
αuαPΛconverges in the strong
ON THE BPBP-NU FOR COMPACT OPERATORS 15
operator topology to the identity on C0pLq˚, and such that PαpC0pLqq is isometrically
isomorphic to a finite-dimensional `8space.
Proof of Theorem 3.4.We will assume first that Lis not compact. Since fjPC0pLq,
j1, . . . , `, there exists a compact set K0ĂLsuch that
sup
j1,...,`
t|fjpxq|:xPLzK0u ă ε
4.
For each xPK0, there exists a relatively compact open subset Uxof Lcontaining xand
such that
|fjpxq ´ fjpyq| ă ε
2for yPUxand j1, . . . , `.
Therefore, tUxuxPK0is a cover of K0, and so, there exist a finite subcover tU1, . . . , UR´1u
such that K0ĎU1Y. . . YUR´1, and if x, y PUrfor some r, then |fjpxq ´ fjpyq| ă ε
2, for
j1, . . . , `.
We define µ:řn
j1|µj| P C0pLq˚. Since for each jP t1, . . . , nuµjis absolutely
continuous with respect to µ, by the Radon-Nikod´ym Theorem, there exists gjPL1pµq
such that µjgjµ, that is,
µjpfq:żL
fdµjżL
fpxqgjpxqdµpxqfor all fPC0pLq.
Since the set of simple functions is dense in L1pµq, we may choose a set of simple functions
tsj:j1, . . . , nusuch that }gj´sj}1ăε
4for j1, . . . , n.
Next, we consider a family tAmuM
m1of pairwise disjoint measurable sets with µpAmq ą 0
for all m, such that each Amis contained in one of the elements of the following cover
of L:tU1, . . . , UR´1, LzK0u, and also tαm,j :m1, . . . , M, j 1, . . . , nusuch that sj
řM
m1αm,j χAm. This cover satisfies that if x, y PLzK0, or if x, y PUr, then |fjpxq ´
fjpyq| ă ε
2for all j1, . . . , ` and all r1, . . . , R ´1. Let Cąmaxt|αm,j |:m
1, . . . , M, j 1, . . . , nu.
Since µis regular, for each 1 ďmďM, we can find a compact set KmĂAmsuch that
µpAmzKmq ă ε
4MC and µpKmq ą 0 for all m1, . . . , M .
Let KK0YK1Y. . . YKM. As KzˆR´1
Ť
r1
Ur˙is a compact subset of L, we can
cover it with finitely many relatively compact open subsets of LzK0that we will denote
UR, UR`1, . . . , UP. If we now apply Lemmas 3.1 and 3.2 to the family tU1, . . . , UPuand
the compacts tK1, . . . , KMuand K, we obtain a refinement of relatively compact open
subsets of L,tZ1, . . . , ZSuwith KmĂZmfor m1, . . . , M and tZ1,...ZMupairwise
disjoint, and defining ZS`1to be the set LzˆS
Ť
s1
Zs˙, we also have a partition of the unity
subordinated to tZsuS
s1,tϕsuS`1
s1, such that:
(i) tϕ1, . . . , ϕMuhave disjoint support.
(ii) ϕmpKmq ” 1 for m1, . . . , M .
(iii) For all MăsďS`1, there exists zsPZssuch that ϕspzsq “ 1.
(iv) For s1, . . . , S `1, supppϕsq Ă Zs.
(v) pϕ1`. . . `ϕSqpKq ” 1.
16 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
Now, we define P:C0pLq ÝÑ C0pLqby
Ppfq:
M
ÿ
m1
1
µpKmqˆżKm
fdµ˙ϕm`
S`1
ÿ
sM`1
fpzsqϕs,for all fPC0pLq.
Let us first check that (2) holds, that is, that }Ppfjq ´ fj} ă εfor all j1, . . . , `. Let
xPL. We will distinguish two cases:
Case 1: if xP
M
Ť
m1
Zm, then there exists exactly one m0such that xPZm0. Then,
for each j1, . . . , `, we have:
|Ppfjqpxq ´ fjpxq| “ ˇˇˇˇˇPpfjqpxq ´
M
ÿ
m1
fjpxqϕmpxq ´
S`1
ÿ
sM`1
fjpxqϕspxqˇˇˇˇˇ
ˇˇˇ
1
µpKm0q˜żKm0
fjpyqdµpyq¸´fjpxqˇˇˇˇˇϕm0pxq
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
(I)
`
`
S`1
ÿ
sM`1
|fjpxq ´ fjpzsq|ϕspxq
looooooooooooooooomooooooooooooooooon
(II)
.
For (I), we have
(I) ˇˇˇˇˇ
1
µpKm0q˜żKm0
pfjpyq ´ fjpxqq dµpyq¸ˇˇˇˇˇϕm0pxq ď 1
µpKm0qżKm0
ε
2dµpyq “ ε
2.
Now, for (II), let sP tM`1, . . . , S `1u. Note that if xRZs, then ϕspxq “ 0,
and if xPZs, we have that |fjpxq ´ fjpzsq| ă ε
2and řS`1
sM`1ϕspxq ď 1, and so,
(II) ăε
2. Therefore, |Ppfjqpxq ´ fjpxq| ă εfor all xP
M
Ť
m1
Zm, for all j1, . . . , `.
Case 2: if xR
M
Ť
m1
Zm, then for each j1, . . . , `, we have
|Ppfjqpxq ´ fjpxq| “ ˇˇˇˇˇ
S`1
ÿ
sM`1
pfjpxq ´ fjpzsqqϕspxqˇˇˇˇˇăε
2
as in item (II) of the previous case.
Summarizing, we get }Ppfjq ´ fj} ă εfor all j1, . . . , `, getting thus (2).
Now we check (1), that is, that }P˚pµjq ´ µj} ă εfor all j1, . . . , n. Indeed, first
observe that if νis a regular Borel (real or complex) measure on L, its associated x˚
νP
C0pLq˚is defined as
x˚
νpfq:żL
fpxqdνpxq,@fPC0pLq,
ON THE BPBP-NU FOR COMPACT OPERATORS 17
and we identify x˚
νν. In our case, we have that
P˚pνqpfq “ żL
Ppfqpxqdνpxq
żL˜M
ÿ
m1
1
µpKmqˆżKm
fdµ˙ϕmpxq¸dνpxq ` żL˜S`1
ÿ
sM`1
fpzsqϕspxq¸dνpxq
M
ÿ
m1
1
µpKmqˆżKm
fdµ˙żL
ϕmpxqdνpxq `
S`1
ÿ
sM`1
fpzsqżL
ϕspxqdνpxq.
In particular, if supppνq Ă
M
Ť
m1
Km, then by Lemma 3.1.(1)
S`1
ÿ
sM`1
fpzsqżL
ϕspxqdνpxq ” 0,@fPC0pLq.
Let now νj:tjµ, where tj:řM
m1αm,j χKm, for all j1, . . . , n, that is,
νjpfq “ żL
fpxq˜M
ÿ
m1
αm,j χKmpxq¸dµpxq,@fPC0pLq.
It holds that P˚pνjq “ νjfor j1, . . . , n. Indeed, as supppνjq Ă
M
Ť
m1
Km, we have
P˚pνjqpfq “
M
ÿ
m1
1
µpKmqˆżKm
fdµ˙żL
ϕmpxq˜M
ÿ
l1
αl,j χKlpxq¸dµpxq
M
ÿ
m1
1
µpKmqˆżKm
fdµ˙żL
αm,j χKmpxqdµpxq
loooooooooooomoooooooooooon
αm,j µpKmq
żL
fpxq˜M
ÿ
m1
αm,j χKmpxq¸dµpxq “ νjpfq
for all fPC0pLqand all j1, . . . , n.
Now, we know that }P˚} “ }P} ď 1 and, since Ppϕjq “ ϕjfor j1, . . . , n, we get that
}P˚} “ 1. Therefore, since P˚pνjq “ νj, we get
}P˚pµjq ´ µj} ď }P˚pµj´νjq} ` }νj´µj}
ď }P˚}¨}µj´νj}`}µj´νj} ď 2}µj´νj}.
But we have
}µj´νj}“}gjµ´tjµ} ď }gjµ´sjµ}`}sjµ´tjµ}
“ }gj´sj}1` }sj´tj}1ăε
4`ε
4ε
2,
18 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
since
}sj´tj}1żLˇˇˇˇˇ
M
ÿ
m1
αm,j χAm´
M
ÿ
m1
αm,j χKmˇˇˇˇˇdµ
ď
M
ÿ
m1
|αm,j |
loomoon
ďC
µpAmzKmq ă MCε
4MC ε
4,
for all j1, . . . , n. Hence,
}P˚pµjq ´ µj} ď 2}µj´νj} ă 2ε
2εfor j1, . . . , n.
Let us finish the proof by checking (3). As µpKmq ą 0, we have Km‰ H,m1, . . . , M.
Hence, we have that zsPZsfor s1, . . . , S `1 and that zs0RŤ
ss0
Zsfor all s0
1, . . . , S `1. By the definition of P, we have that PpC0pLqq “ spantϕs:s1, . . . , S `1u
and we will be done by proving the following equality:
a1ϕ1` ¨ ¨ ¨ ` aS`1ϕS`18maxt|a1|,...,|aS`1|u “ }a}8
for every a“ pa1, . . . , aS`1q. Indeed, for xPL
ˇˇa1ϕ1pxq ` ¨ ¨ ¨ ` aS`1ϕS`1pxqˇˇď }a}8
S`1
ÿ
s1
ϕspxq“}a}8.
But for each s,ˇˇa1ϕ1pzsq ` ¨ ¨ ¨ ` aS`1ϕS`1pzsqˇˇ“ |as|,
and then, a1ϕ1`. . . `aS`1ϕS`18ě }a}8.
Hence, the mapping ρ:`S`1
8ÝÑ C0pLqgiven by
pa1, . . . , aS`1q ÞÝÑ a1ϕ1`. . . `aS`1ϕS`1
is an isometry, and therefore, PpC0pLqq is isometrically isomorphic to `S`1
8.
Now, for the case when Lis compact, by taking K0Land using Lemma 3.3 instead
of Lemma 3.2, a similar proof is valid, except that now all the elements depending on S`1
will vanish in the proof: here we get ZS`1“ H (hence zS`1does not exist), ϕS`10
(and hence it can be omitted), and so, the vector awill only have Scomponents; therefore
PpC0pLqq is isometrically isomorphic to `S
8in this case.
We are now ready to prove the main result of the paper.
Proof of Theorem 1.6.Let f1, . . . , f`PBC0pLq,µ1, . . . , µnPBpC0pLqq˚and εą0 be
given. Let P:C0pLq ÝÑ C0pLqbe the projection from Theorem 3.4, which satisfies that
PpC0pLqq is isometrically isomorphic to `p
8for some pPN. Let r
P:XÝÑ r
PpC0pLqq be
the operator such that r
Ppfq “ Ppfqfor all fPC0pLq, and let i:r
PpC0pLqq ÝÑ C0pLqbe
the natural inclusion. Let ηbe the mapping with which all `n
8spaces has the BPBp-nu for
compact operators (see Corollary 2.4). Since npC0pLqq “ 1 and np`n
pq “ 1 for all nPN(see
[21, Proposition 1.11] for instance), in particular, nkpPpC0pLqqq “ nkpC0pLqq “ 1. There-
fore, we are in the conditions to apply Lemma 2.1 and get that C0pLqhas the BPBp-nu
for compact operators, as desired.
ON THE BPBP-NU FOR COMPACT OPERATORS 19
Alternatively, by Remark 3.5, we may prove Theorem 1.6 applying Proposition 2.2
instead of Lemma 2.1.
Acknowledgment. The authors would like to thank Bill Johnson for kindly answering
several inquiries.
References
[1] M. D. Acosta, Denseness of norm attaining mappings, Rev. R. Acad. Cien. Serie A. Mat. 100
(1-2) (2006), 9-30.
[2] M. D. Acosta, On the Bishop-Phelps-Bollob´as property, Banach Center Publications 119
(2019), 13-32.
[3] M. D. Acosta, R. M. Aron, D. Garc
´
ıa, M. Maestre, The Bishop-Phelps-Bollob´as theorem
for operators, J. Funct. Anal. 294 (2008), 2780-2899.
[4] M. D. Acosta, J. Becerra-Guerrero, Y. S. Choi, M. Ciesielski, S. K. Kim, H. J. Lee,
M. I. Lourenc¸o, M. Mart´
ın, The Bishop-Phelps-Bollob´as property for operators between
spaces of continuous functions, Nonlinear Anal. 95 (2014), 323-332.
[5] M. D. Acosta, M. Fakhar, M. Soleimani-Mourchehkhorti, The Bishop-Phelps-Bollob´as
property for numerical radius of operators on L1pµq,J. Math. Anal. Appl. 458 (2018), 925-936.
[6] R. Aron, Y. S. Choi, S. K. Kim, H. J. Lee, M Mart
´
ın, The Bishop-Phelps-Bollob´as version
of Lindenstrauss properties A and B, Trans. Amer. Math. Soc. 367 (2015), 6085-6101.
[7] A. Avil´
es, A. J. Guirao, J. Rodr
´
ıguez, On the Bishop-Phelps-Bollob´as property for numer-
ical radius in CpKq-spaces, J. Math. Anal. Appl. 419 (2014), 395-421.
[8] E. Bishop, R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math.
Soc. 67 (1961), 97-98.
[9] B. Bollob´
as, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2
(1970), 181-182.
[10] M. Chica, V. Kadets, M. Mart
´
ın, S. Moreno-Pulido, F. Rambla-Barreno, Bishop-
Phelps-Bollob´as moduli of a Banach space J. Math. Anal. Appl. 412 (2014), 697-719.
[11] M. Chica, M. Mart
´
ın, J. Mer
´
ı, Numerical radius of rank-one operators on Banach spaces,
Q. J. Math 65 (2014), 89-100.
[12] Y. S. Choi, S. Dantas, M. Jung, M. Mart´
ın, The Bishop-Phelps-Bollob´as property and
absolute sums, Mediterr. J. Math. 16 (2019), 73, 24 pp.
[13] S. Dantas, D. Garc
´
ıa, M. Maestre, M. Mart
´
ın, The Bishop-Phelps-Bollob´as property for
compact operators, Canad. J. Math. 70 (2018), 56-73.
[14] J. Falc´
o, The Bishop-Phelps-Bollob´as property for numerical radius on L1,J. Math. Anal.
Appl. 414 (1) (2014), 125-133.
[15] I. Gasparis, On contractively complemented subspaces of separable L1-preduals, Israel J. Math.
128 (2002), 77-92.
[16] A. J. Guirao, O. Kozhushkina, The Bishop-Phelps-Bollob´as property for numerical radius
in `1pCq,Studia Math. 218 (2013), 41-54.
[17] J. Johnson, J. Wolfe, Norm attaining operators, Studia Math. 65 (1979), 7-19.
[18] V. Kadets, M. Mart´
ın, R. Pay´
a, Recent progress and open questions on the numerical index
of Banach spaces, RACSAM 100 (2006), 155-182.
[19] V. Kadets, M. Mart´
ın, J. Mer
´
ı, A. P´
erez, A. Quero, On the numerical index with respect
to an operator, Dissertationes Math. 547 (2020), 1–58.
[20] S. K. Kim, H. J. Lee, M. Mart
´
ın, On the Bishop-Phelps-Bollob´as property for numerical
radius, Abstr. Appl. Anal. 2014 (2014), ID 479208.
[21] S. K. Kim, H. J. Lee, M. Mart
´
ın, J. Mer
´
ı, On a second numerical index for Banach spaces,
Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 2, 1003-1051.
[22] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1(1963), 139-148.
[23] M. Mart´
ın, The version for compact operators of Lindenstrauss properties A and B, Rev. R.
Acad. Cien. Serie A. Mat. 110 (2016), 269-284.
[24] M. Mart´
ın, J. Mer
´
ı, and M. Popov, On the numerical index of real Lppµqspaces, Israel J.
Math. 184 (2011), 183-192.
20 D. GARC´
IA, M. MAESTRE, M. MART´
IN, AND O. ROLD´
AN
[25] W. Rudin,Real and Complex Analysis, 3rd Ed., McGraw-Hill, Inc., USA, 1987.
(Domingo Garc´ıa) Departamento de An´
alisis Matem´
atico, Universidad de Valencia, Doctor
Moliner 50, 46100 Burjasot (Valencia), Spain ORCID: 0000-0002-2193-3497
Email address:domingo.garcia@uv.es
(Manuel Maestre) Departamento de An´
alisis Matem´
atico, Universidad de Valencia, Doctor
Moliner 50, 46100 Burjasot (Valencia), Spain ORCID: 0000-0001-5291-6705
Email address:manuel.maestre@uv.es
(Miguel Mart´ın) Departamento de An´
alisis Matem´
atico, Facultad de Ciencias, Universidad
de Granada, 18071 Granada, Spain. ORCID: 0000-0003-4502-798X
Email address:mmartins@ugr.es
(´
Oscar Rold´an) Departamento de An´
alisis Matem´
atico, Universidad de Valencia, Doctor
Moliner 50, 46100 Burjasot (Valencia), Spain. ORCID: 0000-0002-1966-1330
Email address:oscar.roldan@uv.es
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
Article
In this paper, we study conditions assuring that the Bishop–Phelps–Bollobás property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp, (a) if \(Y_1\) is an absolute summand of Y, then \((X,Y_1)\) has the BPBp; (b) if \(X_1\) is an absolute summand of X of type 1 or \(\infty \), then \((X_1,Y)\) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm-attaining operators are also given. We also show that the Bishop–Phelps–Bollobás property for numerical radius is inherited by absolute summands of type 1 or \(\infty \). Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.
Full-text available
Article
We study the Bishop-Phelps-Bollob\'as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikod\'ym property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1< p< \infty$. For $1\leqslant p <\infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(\mu,Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$.
Full-text available
Article
We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces\ldots which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollob\'as property for numerical radius is given.
Full-text available
Article
It has been very recently discovered that there are compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. The aim of this expository paper is to give an overview of those examples and also of sufficient conditions ensuring that compact linear operators can be approximated by norm attaining operators. To do so, we introduce the analogues for compact operators of Lindenstrauss properties A and B.
Full-text available
Article
We study the rank-1 numerical index of a Banach space, namely the infimum of the numerical radii of those rank-1 operators on the space which have norm 1. We show that the rank-1 numerical index is always greater than or equal to 1/e. We also present properties of this index and some examples.
Article
In this paper, we introduce the notion of the Bishop-Phelps-Bollobás property for numerical radius (BPBp-ν) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of L(L1(μ)) have the BPBp-ν for every finite measure μ. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on L1(μ) have the BPBp-ν.
Article
We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(ε) such that for every Y , the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c0-, ℓ1- and ℓ∞-sums of Banach spaces.
Article
We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollob\'as property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an $L_1$-space.