Volume 2009, Article ID 981453,5pages doi:10.1155/2009/981453
Research Article
Denseness of Numerical Radius Attaining
Holomorphic Functions
Han Ju Lee
Department of Mathematics Education, Dongguk University, Seoul 100-715, South Korea
Correspondence should be addressed to Han Ju Lee,hanjulee@dongguk.edu
Received 21 June 2009; Accepted 6 October 2009
Recommended by Nikolaos Papageorgiou
We study the density of numerical radius attaining holomorphic functions on certain Banach spaces using the Lindenstrauss method. In particular, it is shown that if a complex Banach spaceX is locally uniformly convex, then the set of all numerical attaining elements ofABX:Xis dense inABX:X.
Copyrightq2009 Han Ju Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetXbe a complex Banach space andX∗its dual space. We consider the topological subspace ΠX {x, x∗:x∗x 1xx∗}of the product spaceB
X×BX∗, equipped with norm
and weak-∗topology on the unit ballBXofXand its dual unit ballBX∗, respectively. It is easy
to see thatΠXis a closed subspace ofBX×BX∗.
For two complex Banach spacesXandY, denote byCbBX :Ythe Banach space of
all bounded continuous functions fromBXtoY with sup normfsup{fx:x∈BX}. We are interested in the following two subspaces ofCbBX :Y:
AbBX:Y f ∈CbBX:Y:f is holomorphic on the open unit ballB◦X,
AuBX:Y f ∈AbBX :Y:f is uniformly continuous.
1.1
We denote byABX : YeitherAuBX : YorAbBX : Y. WhenY C, we write ABX instead ofABX :C. A nonzero functionf ∈CbBX :Yis said to be astrong peak functionat
x0if whenever there is a sequence{xn}∞n1inBXwith limnfxnf, the sequence{xn}n
ABX. By the maximum modulus theorem, it is easy to see that ifxis a strong peak point of
ABXandX /0, thenxis contained in the unit sphereSXofX.
Harris1introduced the notion of numerical radiusvfof holomorphic functionf∈
AbBX :X. More precisely, for eachf ∈AbBX :X,vf sup{|x∗fx|:x, x∗∈ΠX}. An elementf ∈ABX:Xis said2to be numerical radius attainingif there isx, x∗∈ΠX
such thatvf |x∗fx|.
Acosta and Kim2 showed that if X is a complex Banach space with the Radon-Nikod ´ym property, then the set of all numerical radius attaining elements inABX : Xis
dense. In this paper, we show that ifXis a locally uniformly convex space or locally uniformly
c-convex, order continuous, sequence space, then the set of all numerical radius attaining elements inABX:Xis dense.
We need the notion of numerical boundary. The subsetΓ ofΠXis said3to be a
numerical boundaryofABX:Xif for everyf∈ABX:X,vf sup{|x∗fx|:x, x∗∈Γ}. For more properties of numerical boundaries, see3–5.
2. Main Results
The following is an application of the numerical boundary to the density of numerical radius attaining holomorphic functions. Similar application of the norming subset to the density of norm attaining holomorphic functions is given in4. We use the Lindenstrauss method6.
Theorem 2.1. Suppose thatX is a Banach space and there is a numerical boundaryΓ ⊂ ΠXof
ABX : X such that for every x, x∗ ∈ Γ,x is a strong peak point of ABX. Then the set of
numerical radius attaining elements inABX:Xis dense.
Proof. We may assume thatΓ {xα, x∗α}αandϕαxα 1 for eachα, where eachϕαis a strong peak function inABX. Notice that iff ∈ABX :Xandvf 0, thenvf 0|x∗fx| for anyx, x∗∈ΠXandf attains its numerical radius. Hence we have only to show that iff ∈ ABX : X, vf 1 and > 0, then there isf ∈ ABX : Xsuch thatfattains its
numerical radius andf−f < .
Letf ∈Awithvf 1 andwith 0< <1/3 be given. We choose a monotonically decreasing sequence{k}of positive numbers so that
2
∞
i1
i< , 2
∞
ik 1
i< 2
k, k<
1
10k, k1,2, . . .. 2.1
We next choose inductively sequences{fk}∞k1,{xαk, x∗αk}∞k1inΓsatisfying
f1f, 2.2
x∗
αkfkxαk≥v
fk−2
k, 2.3 fk 1x fkx λkkϕαkx·xαk, 2.4
ϕαkx>1− 1
k impliesx−xαk< 1
where eachλjis chosen inSCto satisfy|x∗αjfjxαj λjj| |x∗αjfjxαj| kand eachϕαj is
ϕnj
αj for some positive integernj. Having chosen these sequences, we verify that the following hold:
fj−fk≤2k−1
ij
i, fk≤f 1
3, j < k, k2,3, . . . , 2.6
vfk 1
≥vfk k−2
k, k1,2, . . . , 2.7 vfk≥vfj, j < k, k2,3, . . . , 2.8
ϕαjxαk>1− 1
j, j < k, k2,3, . . . . 2.9
Assertion2.6is easy by using induction onk. By2.3and2.4,
vfk 1
≥x∗
αkfk 1xαkx∗αkfkxαk λkkϕαkxαk
x∗
αkfkxαk k≥v
fk−2
k k,
2.10
so the relation2.7is proved. Therefore2.8is an immediate consequence2.1and2.7.
Forj < k, by the triangle inequality,2.3and2.6, we have
x∗
αkfj 1xαk≥x∗αkfkxαk−fk−fj 1
≥vfk−2
k−2 k−1
ij 1
i≥vfj 1
−2j2. 2.11
Hence by2.4and2.7,
j ϕαjxαk v
fj≥x∗αkfj 1xαk≥v
fj 1
−22
j
≥vfj j−42j,
2.12
so that
ϕαjxαk≥1−4j>1− 1
j, 2.13
and this proves 2.9. Letf ∈ A be the limit of {fk} in the norm topology. By 2.1and 2.6,f−f limnfn−f1 ≤ 2∞i1i ≤holds. The relations2.5and2.9mean that the sequence {xαk} converges to a pointx, say and by2.3, we havevf limnvfn limn|x∗αnfnxαn| |x∗fx|, wherex∗is a weak-∗limit point of{x∗αk}kinBX∗. Then it is easy
Recall that a Banach spaceXis said to be locally uniformly convexifx∈SXand there is a sequence{xn}inBXsatisfying limnxn x2, then limnxn−x0.
Corollary 2.2. LetX be a locally uniformly convex Banach space. Then the set of numerical radius attaining elements inABX :Xis dense.
Proof. LetΓ ΠXand notice that every element inSX is a strong peak point forAuBX.
Indeed, ifx∈SX, choosex∗∈SX∗so thatx∗x 1. Setfy x∗y 1/2 fory∈BX. Then
f∈ABXandfx 1. If limn|fxn|1 for some sequence{xn}inBX, then limnx∗xn 1.
Since|x∗xn x∗x| ≤ xn x ≤2 for everyn,xn x → 2 andxn−x → 0 asn → ∞. ByTheorem 2.1, we get the desired result.
It was shown in7that if a Banach sequence spaceXis locally uniformlyc-convex and order continuous, then the set of all strong peak points forABXis dense inSX. Therefore, the set of all strong peak points forABXis dense inSX. For the definition of a Banach sequence space and order continuity, see8,9. For the characterization of local uniformc-convexity in function spaces, see7,10.
Corollary 2.3. Suppose that X is a locally uniformly c-convex order continuous Banach sequence space. Then the set of numerical radius attaining elements inAuBX :Xis dense.
Proof. Let Γ {x, x∗ ∈ ΠX : xbe a strong peak point ofAuBX}. Then by 11,
Theorem 2.5, and the remark above theCorollary 2.3,Γis a numerical boundary ofAuBX :
X. Hence the proof is complete byTheorem 2.1.
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and TechnologyNo. 2009-0069100.
References
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