The Bishop-Phelps-Bollobás property for numerical radius in $\ell_1(\mathbb{C})$

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by  Antonio J. Guirao

THE BISHOP-PHELPS-BOLLOBA´S PROPERTY FOR NUMERICAL
RADIUS IN (cid:96) (C)
1
ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
ABSTRACT. We show that the set of bounded linear operators from X to X
admits a Bishop-Phelps-Bolloba´s type theorem for numerical radius whenever
X is(cid:96) (C)orc (C). Asanessentialtoolweprovidetwoconstructiveversions
1 0
3 oftheclassicalBishop-Phelps-Bolloba´stheoremfor(cid:96)1(C).
1
0
2

n 1. INTRODUCTION
a
J The Bishop-Phelps theorem states that norm attaining functionals on a Banach

9 spaceX aredenseinitsdualspaceX∗. In1970, B.Bolloba´sextendedthisresult
1 in a quantitative way in order to work on problems related to the numerical range

ofanoperator[Bol70]. Oneoftheversionsofhisextensionispresentedbelow:
]
A
Theorem 1.1. Let X be a Banach space. Given ε > 0, if x ∈ X, x∗ ∈ X∗ with
F
h. (cid:107)x(cid:107) = (cid:107)x∗(cid:107) = 1 and x∗(x) ≥ 1 − ε22, then there exist elements x0 ∈ X and
x∗ ∈ X∗ suchthat(cid:107)x (cid:107) = (cid:107)x∗(cid:107) = x∗(x ) = 1,
t 0 0 0 0 0
a
m (cid:107)x−x0(cid:107) ≤ εand(cid:107)x∗−x∗0(cid:107) ≤ ε.
[ However, the known proofs of this fact have an existence nature –they are

1 based on Hahn-Banach extension theorem, the Ekeland variational principle or
v Brøndsted-Rockafellar principle. In this paper we construct, as a necessary tool
4 for our main results, explicit expressions of the approximating pair (x ,x∗) when
7 0 0
5 X = (cid:96)1(C)–seeTheorems2.4and2.6.
4 Paralleling the research of norm attaining operators initiated by Lindenstrauss
.
1 in [Lin63], B. Sims raised the question of the norm denseness of the set of nu-
0 mericalradiusattainingoperators–see[Sim72]. Partialpositiveresultshavebeen
3
proved. WeemphasizefortheirimportancetheresultsofM.AcostainherPh. D.
1
: thesis[Aco90],whereasystematicstudyoftheproblemwasinitiated,therenorm-
v
ingresultin[Aco93],andjointfindingsofthisauthorwithR.Paya´ [AP89,AP93].
i
X Priortothem,I.BergandB.Simsin[BS84]gaveapositiveanswerforuniformly
r convex spaces and C. S. Cardassi obtained positive answers for (cid:96) , c , C(K),
a 1 0
L (µ),anduniformlysmoothspaces[Car85a,Car85b,Car85c].
1
Usingarenormingofc ,R.Paya´providedanexampleofaBanachspaceXsuch
0
thatthesetofnumericalradiusattainingoperatorsonX isnotnormdense,answer-
ing in the negative Sims’ question –see [Pay92]. In the same year, M. Acosta, F.
2010MathematicsSubjectClassification. 46B20,47A12.
Keywordsandphrases. Normattaining;Bishop-Phelps-Bolloba´stheorem;numericalradiusat-
tainingoperators.
The research of the first named author was supported in part by MICINN and FEDER
(projectMTM2008-05396),byFundacio´nSe´neca(project08848/PI/08),byGeneralitatValenciana
(GV/2010/036), and by Universidad Polite´cnica de Valencia (project PAID-06-09-2829). The re-
searchofthesecondnamedauthorissupportedbyKentStateUniversity.
1
2 ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
Aguirre, and R. Paya´ in [AAP92] gave another counterexample: X = (cid:96) ⊕ G,
2 ∞
whereGistheGowersspace.
Recently, M. Acosta et al. studied in [AAGM08] a new property, called the
Bishop-Phelps-Bolloba´spropertyforoperators,BPBpforshort. ApairofBanach
spaces (X,Y) has the BPBp if a “Bishop-Phelps-Bolloba´s” type theorem can be
proved for the set of operators from X to Y. This property implies, in particu-
lar, that the norm attaining operators from X to Y are dense in the whole space
of continuous linear operators L(X,Y). However, as shown in [AAGM08], the
converse is not true. Consequently, the BPB property is more than a quantitative
toolforstudyingthedensityofnormattainingoperators.
WeinvestigatehereananalogueoftheBishop-Phelps-Bolloba´spropertyforop-
erators but in relation with numerical radius attaining operators. We call it the
Bishop-Phelps-Bolloba´spropertyfornumericalradius,BPBp-ν forshort. There-
lationbetweennormattainingandnumericalradiusattainingoperatorsisfarfrom
being clear, although the existence of an interconnection is evident. Accordingly,
ourgoalsinthispaperaretodefinethisnewproperty–seeDefinition1below–and
toshowthat(cid:96) (C)andc (C)satisfyit–seeTheorems3.1and 4.1. Thisbringsan
1 0
extensionaswellasaquantitativeversionofC.S.Cardassi’sresultsin[Car85b].
Observe that the counterexamples provided in [AAP92] and [Pay92] imply, in
particular,thatthereexistBanachspacesfailingtheBishop-Phelps-Bolloba´sprop-
ertyfornumericalradius.
GivenaBanachspace(X,(cid:107)·(cid:107)),wedenoteasusualbyS andB ,respectively,
X X
theunitsphereandtheunitballofX. ByX∗ werepresentitsdual,endowedwith
itsstandardnorm(cid:107)x∗(cid:107) = sup {|x∗(x)|}andbyΠ(X)theset
x∈BX
Π(X) = {(x,x∗) ∈ S ×S : x∗(x) = 1}.
X X∗
Givenx ∈ S andx∗ ∈ S ,weset
X X∗
π (x∗) := {x ∈ S : x∗(x) = 1}.
1 X
By L(X) we mean the Banach space of all linear and continuous operators from
X into X endowed with its natural norm (cid:107)T(cid:107) = sup {(cid:107)Tx(cid:107)}. For a given
x∈BX
T ∈ L(X),itsnumericalradiusν(T)isdefinedby
ν(T) = sup{|x∗(Tx)|: (x,x∗) ∈ Π(X)}.
ItiswellknownthatthenumericalradiusofaBanachspaceX isacontinuous
seminorm on X which is, in fact, an equivalent norm when X is complex. In
general,thereexistsaconstantn(X),calledthenumericalindexofX,suchthat
n(X)(cid:107)T(cid:107) ≤ ν(T) ≤ (cid:107)T(cid:107), forallT ∈ L(X).
Our interest in this paper is in spaces of numerical index 1, n(X) = 1, where
the norm and the numerical radius coincide. For background in numerical radius
werefertothemonographs[BD71,BD73]andinnumericalindexwerefertothe
survey[KMP06].
We say that T ∈ L(X) attains its numerical radius if there exists (x,x∗) ∈
Π(X)suchthat|x∗(Tx)| = ν(T). Thesetofnumericalradiusattainingoperators
willbedenotedbyNRA(X) ⊂ L(X).
Definition 1 (BPBp-ν). A Banach space X is said to have the Bishop-Phelps-
Bolloba´spropertyfornumericalradiusifforevery0 < ε < 1, thereexistsδ > 0
such that for a given T ∈ L(X) with ν(T) = 1 and a pair (x,x∗) ∈ Π(X)
THEBISHOP-PHELPS-BOLLOBA´SPROPERTYFORNUMERICALRADIUSIN(cid:96)1(C) 3
satisfying |x∗(Tx)| ≥ 1 − δ, there exist S ∈ L(X) with ν(S) = 1, and a pair
(y,y∗) ∈ Π(X)suchthat
ν(T −S) ≤ ε, (cid:107)x−y(cid:107) ≤ ε, (cid:107)x∗−y∗(cid:107) ≤ ε and |y∗(Sy)| = 1. (1.1)
Observe that if X is a Banach space with n(X) = 1, then the seminorm ν(·)
canbereplacedby(cid:107)·(cid:107)inthedefinitionabove. Notethatallthespacesstudiedin
thispaperhavenumericalindex1.
Notation and terminology. Throughout this paper arg(·) stands for the function
which sends a non zero complex number z to the unique arg(z) ∈ [0,2π) such
that z = |z|earg(z)i. For convenience we extend the function to C by writing
arg(0) = 0. Followingthestandardnotation,letRe(z)andIm(z)be,respectively,
therealandimaginarypartofthecomplexnumberz ∈ C.
All along sections 2 to 4, the spaces (cid:96) , (cid:96) , and c stand respectively for
1 ∞ 0
(cid:96)1(C), (cid:96)∞(C), and c0(C). The standard basis of (cid:96)1 is denoted by {en}n∈N, and
itsbiorthogonalfunctionalsby{e∗n}n∈N. Givenasequenceξ = (ξj)j∈N∈ CN and
acomplexfunctionf: C → Cwewritef(ξ)meaningthesequence(f(ξj))j∈N.
The following sets will be of help in the formulation of the results and proofs.
Givenx = (xj)j∈N ∈ (cid:96)1,ϕ = (ϕj)j∈N ∈ (cid:96)∞ wedefine
N = {j ∈ N: ϕ x = |x |}, (1.2)
(x,ϕ) j j j
supp(x) = {j ∈ N: |x | =(cid:54) 0}.
j
Forr > 0weconsider
A (r) = {j ∈ N: |ϕ | ≥ 1−r}, (1.3)
ϕ j
P (r) = {j ∈ supp(x): Re(ϕ x ) ≥ (1−r)|x |}. (1.4)
(x,ϕ) j j j
ObservethatP (r) ⊂ A (r)andthatifx ≥ 0forallj ∈ N–wedescribethis
(x,ϕ) ϕ j
situationsayingthatxispositive–then
P (r) = {j ∈ supp(x): Re(ϕ ) ≥ (1−r)}.
(x,ϕ) j
For a given set Γ, a subset A ⊂ Γ and K ∈ {R,C}, we denote by 1 the
A
characteristic function of A, that is, the element in KΓ such that (1 ) = 1 if
A γ
γ ∈ Aand(1 ) = 0otherwise.
A γ
2. THE BISHOP-PHELPS-BOLLOBA´S THEOREM IN(cid:96)1(C)
In this section we present two constructive versions of Theorem 1.1, which are
themaintoolintheproofofourTheorems3.1and5.1.
Lemma2.1. Let(x,ϕ) ∈ S ×S . Thenx ∈ π (ϕ)ifandonlyifN = N.
(cid:96)1 (cid:96)∞ 1 (x,ϕ)
Proof. Given a pair (x,ϕ) ∈ S ×S satisfying N = N, one can compute
(cid:96)1 (cid:96)∞ (x,ϕ)
(cid:80) (1.2) (cid:80)
ϕ(x) = ϕ x = |x | = (cid:107)x(cid:107) = 1, which implies that (x,ϕ) ∈
j∈N j j j∈N j
Π((cid:96) ).
1
Conversely,letusassumethat(x,ϕ) ∈ Π((cid:96) )then,
1
(cid:88) (cid:88) (cid:88)
1 = Re(ϕ(x)) = Re(ϕ x ) ≤ |ϕ x | ≤ |x | = 1,
j j j j j
j∈N j∈N j∈N
whichimpliesthatRe(ϕ x ) = |ϕ x | = |x |forj ∈ N. Therefore,ϕ x = |x |
j j j j j j j j
foreveryj ∈ N,whichfinishestheproof. (cid:3)
4 ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
Lemma 2.1 provides the essential insight into the properties of Π((cid:96) ) that we
1
need for the proof of Theorems 2.4 and 2.6. A glance at Lemma 2.1 gives the
followingeasyresultregardingthenormattainingfunctionalson(cid:96) ,NA((cid:96) ).
1 1
Corollary2.2. NA((cid:96) ) = {ϕ ∈ (cid:96) : ∃n ∈ Nwith|ϕ | = (cid:107)ϕ(cid:107)}.
1 ∞ n
The following lemma is an adaptation of [AAGM08, Lemma 3.3] to our nota-
tion.
Lemma 2.3. Let (x,ϕ) ∈ B ×B and 0 < δ < 1 such that ϕ(x) ≥ 1−δ.
Then,foreveryδ < r < 1we(cid:96)h1ave(cid:13)(cid:13)(cid:96)R∞e(cid:0)earg(ϕ)ix(cid:1)·1P (r)(cid:13)(cid:13) ≥ 1−(δ/r).
(x,ϕ)
Proof. Byassumption,wehavethat
1−δ ≤ Re(ϕ(x)) = (cid:88)Re(ϕ x ) = (cid:88)|ϕ |Re(cid:0)earg(ϕj)ix (cid:1)
j j j j
j∈N j∈N
≤ (cid:88) Re(cid:0)earg(ϕj)ix (cid:1)+(1−r) (cid:88) |x |
j j
P (r) N\P (r)
(x,ϕ) (x,ϕ)
≤ r (cid:88) (cid:12)(cid:12)Re(cid:0)earg(ϕj)ix (cid:1)(cid:12)(cid:12)+(1−r),
(cid:12) j (cid:12)
P (r)
(x,ϕ)
whichimpliesthat
(cid:13)(cid:13)Re(cid:0)earg(ϕ)ix(cid:1)1 (cid:13)(cid:13) = (cid:88) (cid:12)(cid:12)Re(cid:16)earg(ϕj)ix (cid:17)(cid:12)(cid:12) ≥ 1−(δ/r),
(cid:13) P(x,ϕ)(r)(cid:13) (cid:12) j (cid:12)
j∈P (r)
(x,ϕ)
aswewantedtoshow. (cid:3)
Observethatthepreviouslemmaimplies,inparticular,that
(cid:13)(cid:13)x·1P (r)(cid:13)(cid:13) ≥ 1−(δ/r).
(x,ϕ)
We present next the two constructive versions of the Bishop-Phelps-Bolloba´s
theorem.
2.1. Firstconstructiveversion.
Theorem2.4. Given(x, ϕ) ∈ B ×B and0 < ε < 1suchthatϕ(x) ≥ 1−ε3.
(cid:96)1 (cid:96)∞ 4
Then, there exists (x ,ϕ ) ∈ Π((cid:96) ) such that (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε.
0 0 1 0 0
Moreover,wecantake
x0 := (cid:13)(cid:13)x·1P (ε2/2)(cid:13)(cid:13)−1·x·1P (ε2/2). (2.1)
(x,ϕ) (x,ϕ)
Proof. Set P := P (ε2/2) –see definition (1.4). Applying Lemma 2.3 with
(x,ϕ)
δ = ε2/2andr = εgivesthat
M:=(cid:107)x·1 (cid:107) ≥ 1−(ε/2). (2.2)
P
Letusdefine
ϕ0 := ϕ·1N\P +e−arg(x)i·1P ∈ S(cid:96)∞ (2.3)
and
x := M−1x·1 ∈ S . (2.4)
0 P (cid:96)1
Ononehand,wecancompute
(cid:107)x−x0(cid:107) (2=.4) (cid:13)(cid:13)x−M−1x·1P(cid:13)(cid:13) = (M−1−1)(cid:107)x·1P(cid:107)+(cid:13)(cid:13)x·1N\P(cid:13)(cid:13)
(2=.2) (1−M)+(cid:13)(cid:13)x·1N\P(cid:13)(cid:13)(cid:107)x≤(cid:107)≤12−2M(2≤.2)ε,
THEBISHOP-PHELPS-BOLLOBA´SPROPERTYFORNUMERICALRADIUSIN(cid:96)1(C) 5
and, since the support of x is included in P –this is a consequence of (2.4), we
0
deducethat
ϕ (x ) = (cid:88)(ϕ ) (x ) (2=.3) (cid:88)e−arg(xj)i(x ) (2=.4) (cid:88)|(x ) |= (cid:107)x (cid:107) = 1,
0 0 0 j 0 j 0 j 0 j 0
j∈P j∈P j∈P
whichisequivalentlyexpressedas(x ,ϕ ) ∈ Π((cid:96) ).
0 0 1
Ontheotherhand,usingthat
(cid:112)
|z−1| ≤ 2(1−Re(z))foreveryz ∈ Csuchthat|z| ≤ 1, (2.5)
wededuce
(cid:107)ϕ−ϕ0(cid:107) (2=.3) sup{|ϕj −(ϕ0)j|} (2=.3) sup(cid:8)(cid:12)(cid:12)ϕj −e−arg(xj)i(cid:12)(cid:12)(cid:9)
j∈P j∈P
= sup(cid:8)(cid:12)(cid:12)earg(xj)iϕj −1(cid:12)(cid:12)(cid:9) (2≤.5) sup(cid:26)(cid:113)2−2Re(cid:0)earg(xj)iϕj(cid:1)(cid:27)
j∈P j∈P
(cid:112)
≤ 2−2(1−ε2/2) = ε,
whichfinishestheproof. (cid:3)
An immediate consequence of Theorem 2.4 is the following version of the
Bishop-Phelps-Bolloba´stheoremfor(cid:96) (C).
1
Corollary2.5. Let0 < ε < 1and(x, ϕ) ∈ B ×B suchthat|ϕ(x)| ≥ 1−ε3.
(cid:96)1 (cid:96)∞ 4
Then, there exists (x ,ϕ ) ∈ S ×S such that (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε
0 0 (cid:96)1 (cid:96)∞ 0 0
and|ϕ (x )| = 1.
0 0
Proof. Apply Theorem 2.4 to the pair (cid:0)e−arg(ϕ(x))ix,ϕ(cid:1) obtaining (z ,ϕ ) be-
0 0
(cid:13) (cid:13)
longing to Π((cid:96)1) such that (cid:13)e−arg(ϕ(x))ix−z0(cid:13) ≤ ε and (cid:107)ϕ−ϕ0(cid:107) ≤ ε. There-
fore, if we set x :=earg(ϕ(x))iz , the pair (x ,ϕ ) satisfies the conclusions of the
0 0 0 0
corollary. (cid:3)
2.2. Second constructive version. Given a pair (x,ϕ) and 0 < ε < 1, The-
orem 2.4 ensures the existence of a pair (x ,ϕ ) –defined by (2.4) and (2.3)–
0 0
satisfying the conclusions of the Bishop-Phelps-Bolloba´s theorem. However, ϕ
0
depends on x, in fact, on arg(x). In order to prove Theorem 3.1 we will need a
functional ϕ depending only on the given ε and ϕ. So, we present the following
0
result.
Theorem2.6. Let(x, ϕ) ∈ B ×B and 0 < ε < 1besuchthatϕ(x) ≥ 1−ε3.
(cid:96)1 (cid:96)∞ 60
Then there exists (x ,ϕ ) ∈ Π((cid:96) ) such that (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε.
0 0 1 0 0
Moreover,thefunctionalϕ canbedefinedas
0
ϕ0 = ϕ·1N\Aϕ(ε2/20)+earg(ϕ)i·1Aϕ(ε2/20). (2.6)
Proof. LetusconsidertheisometryS: (cid:96) → (cid:96) definedby
1 1
(cid:104)e∗,Sy(cid:105) = earg(ϕj)iy , fory ∈ (cid:96) and j ∈ N. (2.7)
j j 1
Setx = Sxandϕ = ϕ◦S−1. Then,itisclearthatthepair(x,ϕ)isinB ×B ,
(cid:101) (cid:101) (cid:101) (cid:101) (cid:96)1 (cid:96)∞
that ϕ(cid:101)(x(cid:101)) ≥ 1 − 6ε30 and that ϕ(cid:101) = (|ϕj|)j∈N is positive. Denote by A and P
respectively the sets A (r) and P (r) –see definitions (1.3) and (1.4), where
ϕ (x,ϕ)
(cid:101) (cid:101) (cid:101)
r:=ε2. Letusdefine
20
ϕ(cid:98):=ϕ(cid:101)·1N\A+1A ∈ S(cid:96)∞ (2.8)
6 ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
and
x:=M−1Re(x)·1 ∈ S , (2.9)
(cid:98) (cid:101) P (cid:96)1
whereM:=(cid:107)Re(x)·1 (cid:107). ApplyingLemma2.3withδ = ε3/60andr,givesthat
(cid:101) P
M ≥ 1− ε. Inparticular,thismeansthatP,andthusA,arenon-empty.
3
Wecancomputethat
(2.8) (2.8)
(cid:107)ϕ−ϕ(cid:107) = sup{|ϕ −ϕ |} = sup{|ϕ −1|}
(cid:101) (cid:98) (cid:101)j (cid:98)j (cid:101)j
j∈A j∈A
(1.3)
= sup{(1−ϕ )} ≤ r ≤ ε, (2.10)
(cid:101)j
j∈A
and, since by (1.4) and (2.9) the support of x is P ⊂ A –which, in particular,
(cid:98)
impliesthatx > 0forj ∈ P,wededucethat
(cid:98)j
(cid:88) (2.8) (cid:88) (cid:88)
ϕ(x) = ϕ x = x = |x | = 1, (2.11)
(cid:98) (cid:98) (cid:98)j(cid:98)j (cid:98)j (cid:98)j
j∈P j∈P j∈P
whichisequivalentlywrittenas(x,ϕ) ∈ Π((cid:96) ).
(cid:98) (cid:98) 1
Inordertoshowthat(cid:107)x−x(cid:107) ≤ ε,letusobservefirstthat
(cid:101) (cid:98)
(cid:88) (cid:88) ε
(cid:107)x·1 (cid:107) = |x | ≥ |Re(x )| =M ≥ 1− , (2.12)
(cid:101) P (cid:101)j (cid:101)j
3
j∈P j∈P
fromwhich
(cid:107)x(cid:101)−x(cid:98)(cid:107) (2=.9) (cid:13)(cid:13)x(cid:101)−M−1Re(x(cid:101))·1P(cid:13)(cid:13) = (cid:13)(cid:13)x(cid:101)·1N\P(cid:13)(cid:13)+(cid:13)(cid:13)(x(cid:101)−M−1Re(x(cid:101)))·1P(cid:13)(cid:13)
(2≤.12) ε +(cid:13)(cid:13)(x(cid:101)−M−1Re(x(cid:101)))·1P(cid:13)(cid:13). (2.13)
3
Weneedabitmorecaretoestimatethelasttermin(2.13). Fromtheverydefinition
ofP,weknowthatforeveryj ∈ P itholds
|x | ≤ (1−r)−1ϕ Re(x ). (2.14)
(cid:101)j (cid:101)j (cid:101)j
Therefore,
(cid:88) (cid:88)
(cid:107)(x−Re(x))·1 (cid:107) = |x −Re(x )| = |Im(x )|
(cid:101) (cid:101) P (cid:101)j (cid:101)j (cid:101)j
j∈P j∈P
(cid:88)(cid:113)
= |x |2−Re(x )2
(cid:101)j (cid:101)j
j∈P
(2.14) (cid:88) (cid:112)
≤ |Re(x )| (1−r)−2−1
(cid:101)j
j∈P
(cid:112) r=ε220 ε
≤ (cid:107)x(cid:107) (1−r)−2−1 ≤ , (2.15)
(cid:101)
3
whichimpliesthat
(cid:13)(cid:13)(x(cid:101)−M−1Re(x(cid:101)))·1P(cid:13)(cid:13) ≤ (cid:107)(x(cid:101)−Re(x(cid:101)))·1P(cid:107)+(cid:13)(cid:13)(1−M−1)Re(x(cid:101))·1P(cid:13)(cid:13)
(2.15) ε
≤ +(M−1−1)(cid:107)Re(x)·1 (cid:107)
(cid:101) P
3
ε 2ε
= +(1−M) ≤ . (2.16)
3 3
THEBISHOP-PHELPS-BOLLOBA´SPROPERTYFORNUMERICALRADIUSIN(cid:96)1(C) 7
Puttingtogether(2.13)and(2.16),oneobtains
(cid:107)x(cid:101)−x(cid:98)(cid:107) ≤ ε +(cid:13)(cid:13)(x(cid:101)−M−1Re(x(cid:101)))·1P(cid:13)(cid:13) ≤ ε, (2.17)
3
whichfinishesthecoreoftheproof.
Now,wedefine
x :=S−1x and ϕ = S∗(ϕ) = ϕ◦S, (2.18)
0 (cid:98) 0 (cid:98) (cid:98)
whichby(2.11)givesthatϕ (x ) = ϕ(x) = 1. SinceS andS∗ areisometries,we
0 0 (cid:98) (cid:98)
deducefrom(2.10),(2.17),(2.18)andthedefinitionofxandϕthat
(cid:101) (cid:101)
(cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε.
0 0
Therefore,(x ,ϕ )isthepairinΠ((cid:96) )wewerelookingfor.
0 0 1
Bearinginmind(2.18),onecomputes
(ϕ ) = ϕ (e ) (2=.18) ϕ(Se ) (2=.7) ϕ(cid:0)earg(ϕj)ie (cid:1) = earg(ϕj)iϕ ,
0 j 0 j (cid:98) j (cid:98) j (cid:98)j
whichtogetherwith(2.8)impliesthatϕ0 = ϕ·1N\A+earg(ϕ)i·1A. Finally,noting
thatA = A (r) = A (r),thevalidityof(2.6)hasbeenshown. (cid:3)
ϕ ϕ
(cid:101)
Remark 2.7. Observe that the function ϕ provided by Theorem 2.6 and defined
0
by(2.6)onlydependsonεandϕitselfaswellassatisfiesπ (ϕ) ⊂ π (ϕ ).
1 1 0
3. BPB PROPERTY FOR NUMERICAL RADIUS IN(cid:96)1(C)
As a consequence of Theorems 2.4 and 2.6 we show that (cid:96) has the Bishop-
1
Phelps-Bolloba´spropertyfornumericalradius.
Theorem3.1. LetT ∈ S ,0 < ε < 1and(x,ϕ) ∈ Π((cid:96) )suchthatϕ(Tx) ≥
L((cid:96)1) 1
1−(ε/9)9/2. ThenthereexistT ∈ S and(x ,ϕ ) ∈ Π((cid:96) )suchthat
0 L((cid:96)1) 0 0 1
(cid:107)T −T (cid:107) ≤ ε, (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε and ϕ (T x ) = 1. (3.1)
0 0 0 0 0 0
(cid:112)
Proof. First of all, fix µ := ε3/240. Using a suitable isometry, we can assume
that x is positive. In particular, by Lemma 2.1 and the definition of N in (1.2),
x,ϕ
wecanassumethatϕ = 1forj ∈ supp(x). Sinceµ3/4 ≥ (ε/9)9/2,Theorem2.4
j
can be applied to the pair (x, T∗ϕ) ∈ B ×B and µ instead of ε giving x ∈
(cid:96)1 (cid:96)∞ 0
π (ϕ)suchthat(cid:107)x−x (cid:107) ≤ µ ≤ ε. Moreover,by(2.1)weknowthat
1 0
x = (cid:107)x·1 (cid:107)−1·x·1 , (3.2)
0 P P
wherethenon-emptysetP isdefinedby
P := P (µ2/2) = {j ∈ supp(x): Re(T∗ϕ(e )) ≥ 1−µ2/2}. (3.3)
(x,T∗ϕ) j
Inparticular,x ispositive.
0
Since µ2/2 = (ε/2)3, for each j ∈ P we can apply Theorem 2.6 to the pair
60
(e−arg(ϕ(Tej))iTej,ϕ)andε/2tofind(zj,ϕ0) ∈ Π((cid:96)1)suchthat
(cid:107)Te −a z (cid:107) ≤ ε/2, (cid:107)ϕ−ϕ (cid:107) ≤ ε/2
j j j 0
andΠ1(ϕ) ⊂ Π1(ϕ0)–seeRemark2.7,whereaj = earg(ϕ(Tej))i. Observethatϕ0
canbechosenindependentlyonj ∈ P andby(2.6)explicitlywrittenas
ϕ0 = ϕ·1N\Aϕ(ε2/80)+earg(ϕ)i·1Aϕ(ε2/80). (3.4)
8 ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
LetusdefineT astheuniqueoperatorinL((cid:96) )suchthatT e = Te fori ∈/ P
0 1 0 i i
andT e = z forj ∈ P. Equivalently,
0 j j
(cid:88)
T0x = 1N\P ·Tx+ e∗j(x)zj, forx ∈ (cid:96)1. (3.5)
j∈P
Itisclearfrom(3.5)that
(cid:26) (cid:27)
(cid:107)T (cid:107) = sup{(cid:107)T e (cid:107)} = max sup{(cid:107)Te (cid:107)}, sup{(cid:107)z (cid:107)} = 1.
0 0 n j j
n∈N j∈/P j∈P
Givenj ∈ P,theidentity(3.3)ensuresthatRe(ϕ(Te )) ≥ 1−µ2/2. Usingagain
j
thegeneralfact(2.5),wededucethat|a −1| ≤ µ ≤ ε/2.
j
Therefore,
(cid:107)T −T (cid:107) = sup{(cid:107)Te −T e (cid:107)} = sup{(cid:107)Te −z (cid:107)}
0 n 0 n j j
n∈N j∈P
≤ sup{(cid:107)Te −a z (cid:107)}+sup{(cid:107)a z −z (cid:107)}
j j j j j j
j∈P j∈P
ε
≤ +sup{|a −1|} ≤ ε.
j
2
j∈P
Since x ∈ π (ϕ) and π (ϕ) ⊂ π (ϕ ), we deduce that (x ,ϕ ) belongs to
0 1 1 1 0 0 0
Π((cid:96) ). It remains to show that ϕ (T x ) = 1 to prove the validity of (3.1). But,
1 0 0 0
sincex ispositive,weobtainthat
0
(3.5) (cid:88) (cid:88)
ϕ (T x ) = (x ) ϕ (z )+ (x ) ϕ (Te )
0 0 0 0 j 0 j 0 j 0 j
j∈P j∈/P
(3.2) (cid:88) (cid:88)
= (x ) = |(x ) | = (cid:107)x (cid:107) = 1,
0 j 0 j 0
j∈P j∈P
andtheproofisover. (cid:3)
Remark3.2. Wecannotreplacethecondition(x,ϕ) ∈ Π((cid:96) )inTheorem3.1bythe
1
moregeneral(x,ϕ) ∈ B ×B . Indeed,letusconsidertheoperatorT: (cid:96) → (cid:96)
(cid:96)1 (cid:96)∞ 1 1
defined by Te = e for j ≥ 2 and Te = e . Take (e ,e∗) ∈ B × B ,
j j 1 2 1 2 (cid:96)1 (cid:96)∞
T ∈ L((cid:96) ),and(x,ϕ) ∈ B ×B suchthat(cid:107)T −T (cid:107) ≤ ε,(cid:107)e −x(cid:107) ≤ ε,and
0 1 (cid:96)1 (cid:96)∞ 0 1
(cid:107)e∗−ϕ(cid:107) ≤ ε. Then
2
|ϕ(x)| ≤ |ϕ(x)−e∗(x)|+|e∗(x)−e∗(e )|+|e∗(e )| ≤ 2ε,
2 2 2 1 2 1
whichimpliesthat(x,ϕ)cannotbeinΠ((cid:96) ).
1
Corollary3.3. TheBanachspace(cid:96) hastheBishop-Phelps-Bolloba´spropertyfor
1
numericalradius.
Proof. Let us consider T ∈ L((cid:96) ) with ν(T) = 1 and 0 < ε < 1. Let us take a
1
9
pair(x,ϕ) ∈ Π((cid:96)1)suchthat|ϕ(Tx)| ≥ 1−(ε/9)2. Infact,wecanassumethat
ϕ(Tx) ≥ 1−(ε/9)92;otherwise,weproceedwithT(cid:101) = e−arg(ϕ(Tx))iT. ThenThe-
orem3.1givestheexistenceofanoperatorT ∈ S andapair(x ,ϕ )∈ Π((cid:96) )
0 L((cid:96)1) 0 0 1
thatsatisfyconditionsin(3.1),whicharepreciselytherequirements(1.1)inDefi-
nition1. (cid:3)
Corollary3.4([Car85b]). ThesetNRA((cid:96) )isdenseinL((cid:96) ).
1 1
THEBISHOP-PHELPS-BOLLOBA´SPROPERTYFORNUMERICALRADIUSIN(cid:96)1(C) 9
4. BPB PROPERTY FOR NUMERICAL RADIUS INc0(C)
Theorem3.1allowsustoshowthatc hastheBishop-Phelps-Bolloba´sproperty
0
for numerical radius as well. Indeed, we rely on the fact that our constructions in
(cid:96) canbedualized.
1
Theorem4.1. LetT ∈ S ,0 < ε < 1and(x,ϕ) ∈ Π(c )suchthat|ϕ(Tx)| ≥
L(c0) 0
1−(ε/9)9/2. ThenthereexistS ∈ S and(x ,ϕ ) ∈ Π(c ),suchthat
L(c0) 0 0 0
(cid:107)T −S(cid:107) ≤ ε, (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε and ϕ (Sx ) = 1.
0 0 0 0
Proof. Throughoutthisproofweidentifytheelementsinc withtheirimagein(cid:96)
0 ∞
throughthenaturalembeddingc → (cid:96) . TheadjointoperatorofT,T∗: (cid:96) → (cid:96)
0 ∞ 1 1
satisfies
|x(T∗ϕ)| = |T∗(ϕ)(x)| = |ϕ(Tx)|≥1−(ε/9)9/2.
Without loss of generality, we can assume that x(T∗ϕ) ≥ 1−(ε/9)9/2. Oth-
erwise, employingtechniquesfromtheproofofCorollary3.3, definetheoperator
T(cid:101) = e−arg(x(T∗ϕ))iT∗ andproceedwiththeproofforx(T(cid:101)ϕ) = |x(T∗ϕ)|.
ByTheorem3.1,thereexistsT ∈ L((cid:96) ),(cid:107)T (cid:107) = 1and(ϕ ,x ) ∈ Π((cid:96) )such
0 1 0 0 0 1
that
(cid:107)T∗−T (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε, (cid:107)x−x (cid:107) ≤ ε
0 0 0
andx (T ϕ ) = 1.
0 0 0
We assert that (x ,ϕ ) is the pair we are looking for. To show this, we will
0 0
reexamine the proof of Theorem 3.1 to establish how x , ϕ and T are defined.
0 0 0
Indeed,from(3.3),(3.2),(3.4)and(3.5)wehaverespectively
P = P (ε3/480),
(ϕ,T∗∗x)
ϕ = (cid:107)ϕ·1 (cid:107)−1·ϕ·1 ,
0 P P
x0 = x·1N\Ax(ε2/80)+earg(x)i·1Ax(ε2/80), (4.1)
(cid:88)
T0x = 1N\P ·Tx+ e∗j(x)zj, forx ∈ (cid:96)1,
j∈P
where{z } ⊂ π (ϕ ).
j j∈P 1 0
Note that A (ε2/80) = {j ∈ N: |x | ≥ 1−ε2/80} and that x ∈ c . Thus,
x j 0
A (ε2/80)isfinitewhich,by(4.1),impliesthatx ∈ c .
x 0 0
WeshallshowthatT isanadjointoperatorandthusthatthereexistsS ∈ L(c )
0 0
suchthatS∗ = T . ItwillbeenoughtoshowthatT∗| ⊂ c . Sett = (cid:104)e ,T(e )(cid:105)
0 0 c0 0 ij i j
fori,j ∈ N. Fixi ∈ N,thenforj ∈ N
(cid:26)
t ifj ∈/ P,
(cid:104)e ,T∗(e )(cid:105) = ji
j 0 i (z ) ifj ∈ P.
j i
Since x ∈ c , T∗∗x belongs to c , which implies that P is finite. Accordingly,
0 0
onlyfinitelymanytermsoftheform(cid:104)e ,T∗(e )(cid:105)differfromthecorrespondingt .
j 0 i ji
Ontheotherhand,sinceT belongstoL(c ),itholdsthatlim |t | = 0. Therefore,
0 j ji
wededucethat|(cid:104)e ,T∗(e )(cid:105)| → 0whenj → ∞. ThisimpliesthatT∗e ∈ c and,
j 0 i 0 i 0
sincei ∈ Nisarbitrarilychosen,wededucethatT∗| ⊂ c .
0 c0 0
HenceweobtaintheoperatorS = T∗| ∈ L(c )andthepair(x ,ϕ ) ∈ Π(c )
0 c0 0 0 0 0
satisfying:
ϕ (Sx ) = S∗ϕ (x ) = x (S∗ϕ ) = x (T ϕ ) = 1,
0 0 0 0 0 0 0 0 0
10 ANTONIOJ.GUIRAOANDOLENAKOZHUSHKINA
and
(cid:107)S −T(cid:107) = (cid:107)(S −T)∗(cid:107) = (cid:107)S∗−T∗(cid:107) = (cid:107)T −T∗(cid:107) ≤ ε,
0
whichfinishestheproof. (cid:3)
Theorem4.1impliesthefollowingtwocorollaries.
Corollary4.2. TheBanachspacec hastheBishop-Phelps-Bolloba´spropertyfor
0
numericalradius.
Corollary4.3([Car85b]). ThesetNRA(c )isdenseinL(c ).
0 0
5. GENERALIZATIONS AND REMARKS
All the results that have been presented in sections 2, 3 and 4 were stated and
proved for the Banach spaces (cid:96) (C) or c (C). However, a glance at their proofs
1 0
suffices to convince oneself of their validity for (cid:96) (R) and c (R) –shorter proofs
1 0
and better estimates can be obtained in this case. More generally, given a non-
emptysetΓandK ∈ {R,C},theseresultsare,aftersuitableadjustments,stillvalid
for (cid:96) (Γ,K) and c (Γ,K). The spaces (cid:96) (Γ,K) and c (Γ,K) are, respectively,
1 0 1 0
the (cid:96) -sum and the c -sum of Γ copies of the field K. Note that in particular
1 0
(cid:96) (N,K) = (cid:96) (K).
1 1
TheBanachspacec (Γ,K)isapredualof(cid:96) (Γ,K). Observethatbothc (Γ,K)
0 1 0
and (cid:96) (Γ,K) have numerical index 1. Previous considerations imply that both
1
of them also have the BPB property for numerical radius. The ω∗ topology of
(cid:96) (Γ,K) stands here for the topology induced on (cid:96) (Γ,K) by pointwise conver-
1 1
genceonelementsofc (Γ,K).
0
On the other hand, the proof of Theorem 4.1 shows that in Theorem 3.1 we
proved more than was stated. Indeed, putting together Theorem 3.1, the ideas on
duality in the proof of Theorem 4.1 and considerations above, one easily proves
thefollowingtheorem.
Theorem 5.1. Let T ∈ SL((cid:96)1(Γ,K)), 0 < ε < 1 and (x,ϕ) ∈ Π((cid:96)1(Γ,K)) such
that ϕ(Tx) ≥ 1 − (ε/9)9/2. Then there exist T ∈ S and (x ,ϕ ) ∈
0 L((cid:96)1(Γ,K)) 0 0
Π((cid:96) (Γ,K))suchthat
1
(cid:107)T −T (cid:107) ≤ ε, (cid:107)x−x (cid:107) ≤ ε, (cid:107)ϕ−ϕ (cid:107) ≤ ε and ϕ (T x ) = 1.
0 0 0 0 0 0
Moreover,ifT isω∗-ω∗-continuousandϕisω∗-continuous,thenT andϕ will
0 0
beω∗-ω∗-continuousandω∗-continuous,respectively.
BelowaretwoconsequencesofTheorem5.1.
Theorem 5.2. The Banach space (cid:96) (Γ,K) has the BPB property for numerical
1
radius.
Theorem 5.3. The Banach space c (Γ,K) has the BPB property for numerical
0
radius.
Proof. Fix 0 < ε < 1, δ ≤ (ε/9)9/2, T ∈ SL(c0(Γ,K)) and (x,x∗) ∈ Π(c0(Γ,K))
suchthatx∗(Tx)≥1−δ. ApplyingTheorem5.1totheω∗-ω∗-continuousoperator
T∗ ∈ SL((cid:96)1(Γ,K)), the pair (x∗,x) and ε, gives a new T0 ∈ SL(c0(Γ,K)) and a new
pair(x∗,x∗∗) ∈ Π((cid:96) (Γ,K))satisfying
0 0 1
(cid:13)(cid:13)T∗−T∗(cid:13)(cid:13) ≤ ε, (cid:107)x−x∗∗(cid:107) ≤ ε, (cid:107)x∗−x∗(cid:107) ≤ εandx∗∗(cid:0)T∗x∗(cid:1) = 1. (5.1)
0 0 0 0 0 0