- Research Article
- Open Access

# Denseness of Numerical Radius Attaining Holomorphic Functions

- Han Ju Lee
^{1}Email author

**2009**:981453

https://doi.org/10.1155/2009/981453

© Han Ju Lee. 2009

**Received:**21 June 2009**Accepted:**6 October 2009**Published:**8 October 2009

## Abstract

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 space is locally uniformly convex, then the set of all numerical attaining elements of is dense in .

## Keywords

- Banach Space
- Positive Integer
- Continuous Function
- Function Space
- Holomorphic Function

## 1. Introduction

Let be a complex Banach space and its dual space. We consider the topological subspace of the product space , equipped with norm and weak- topology on the unit ball of and its dual unit ball , respectively. It is easy to see that is a closed subspace of .

We denote by
either
or
. When
, we write
instead of
. A nonzero function
is said to be a *strong peak function* at
if whenever there is a sequence
in
with
, the sequence
converges to
. The corresponding point
is said to be a *strong peak point* of
. It is easy to see that
is a strong peak point of
if and only if
is a strong peak point of
. By the maximum modulus theorem, it is easy to see that if
is a strong peak point of
and
, then
is contained in the unit sphere
of
.

Harris [1] introduced the notion of numerical radius
of holomorphic function
. More precisely, for each
,
. An element
is said [2] to be *numerical radius attaining* if there is
such that
.

Acosta and Kim [2] showed that if is a complex Banach space with the Radon-Nikodým property, then the set of all numerical radius attaining elements in is dense. In this paper, we show that if is a locally uniformly convex space or locally uniformly -convex, order continuous, sequence space, then the set of all numerical radius attaining elements in is dense.

We need the notion of numerical boundary. The subset
of
is said [3] to be a *numerical boundary* of
if for every
,
. For more properties of numerical boundaries, see [3–5].

## 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 in [4]. We use the Lindenstrauss method [6].

Theorem 2.1.

Suppose that is a Banach space and there is a numerical boundary of such that for every , is a strong peak point of . Then the set of numerical radius attaining elements in is dense.

Proof.

We may assume that and for each , where each is a strong peak function in . Notice that if and , then for any and attains its numerical radius. Hence we have only to show that if and , then there is such that attains its numerical radius and

and this proves (2.9). Let be the limit of in the norm topology. By (2.1) and (2.6), holds. The relations (2.5) and (2.9) mean that the sequence converges to a point , say and by (2.3), we have , where is a weak- limit point of in . Then it is easy to see that . Hence attains its numerical radius. This concludes the proof.

Recall that a Banach space
is said to be *locally uniformly convex* if
and there is a sequence
in
satisfying
, then
.

Corollary 2.2.

Let be a locally uniformly convex Banach space. Then the set of numerical radius attaining elements in is dense.

Proof.

Let and notice that every element in is a strong peak point for . Indeed, if , choose so that . Set for . Then and . If for some sequence in , then . Since for every , and as . By Theorem 2.1, we get the desired result.

It was shown in [7] that if a Banach sequence space is locally uniformly -convex and order continuous, then the set of all strong peak points for is dense in . Therefore, the set of all strong peak points for is dense in . For the definition of a Banach sequence space and order continuity, see [8, 9]. For the characterization of local uniform -convexity in function spaces, see [7, 10].

Corollary 2.3.

Suppose that is a locally uniformly -convex order continuous Banach sequence space. Then the set of numerical radius attaining elements in is dense.

Proof.

Let . Then by [11, Theorem 2.5], and the remark above the Corollary 2.3, is a numerical boundary of . Hence the proof is complete by Theorem 2.1.

## Declarations

### 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 Technology (No. 2009-0069100).

## Authors’ Affiliations

## References

- Harris LA:
**The numerical range of holomorphic functions in Banach spaces.***American Journal of Mathematics*1971,**93:**1005–1019. 10.2307/2373743MathSciNetView ArticleMATHGoogle Scholar - Acosta MD, Kim SG:
**Denseness of holomorphic functions attaining their numerical radii.***Israel Journal of Mathematics*2007,**161:**373–386. 10.1007/s11856-007-0083-xMathSciNetView ArticleMATHGoogle Scholar - Acosta MD, Kim SG:
**Numerical Shilov boundaries for some classical Banach spaces.***Journal of Mathematical Analysis and Applications*2009,**350**(2):694–707. 10.1016/j.jmaa.2008.05.064MathSciNetView ArticleMATHGoogle Scholar - Choi YS, Lee HJ, Song HG: Bishop's theorem and differentiability of a sub-space of . to appear in Israel Journal of Mathematics to appear in Israel Journal of MathematicsGoogle Scholar
- Kim SG, Lee HJ: Numerical peak holomorphic functions on Banach spaces. Journal of Mathematical Analysis and Applications, to appear Journal of Mathematical Analysis and Applications, to appearGoogle Scholar
- Lindenstrauss J:
**On operators which attain their norm.***Israel Journal of Mathematics*1963,**1:**139–148. 10.1007/BF02759700MathSciNetView ArticleMATHGoogle Scholar - Choi YS, Han KH, Lee HJ:
**Boundaries for algebras of holomorphic functions on Banach spaces.***Illinois Journal of Mathematics*2007,**51**(3):883–896.MathSciNetMATHGoogle Scholar - Lee HJ:
**Complex convexity and monotonicity in quasi-Banach lattices.***Israel Journal of Mathematics*2007,**159:**57–91. 10.1007/s11856-007-0038-2MathSciNetView ArticleMATHGoogle Scholar - Lindenstrauss J, Tzafriri L:
*Classical Banach Spaces II*.*Volume 97*. Springer, Berlin, Germany; 1979.View ArticleMATHGoogle Scholar - Lee HJ: Randomized series and geometry of Banach spaces. to appear in Taiwanese Journal of Mathematics to appear in Taiwanese Journal of MathematicsGoogle Scholar
- Rodríguez-Palacios Á:
**Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space.***Journal of Mathematical Analysis and Applications*2004,**297**(2):472–476. 10.1016/j.jmaa.2004.03.012MathSciNetView ArticleMATHGoogle Scholar

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