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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 380475, 8 pages
http://dx.doi.org/10.1155/2013/380475
Research Article

Generalized Numerical Index and Denseness of Numerical Peak Holomorphic Functions on a Banach Space

1Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Mathematics Education, Dongguk University-Seoul, Seoul 100-715, Republic of Korea

Received 11 June 2013; Accepted 23 August 2013

Academic Editor: Geraldo Botelho

Copyright © 2013 Sung Guen Kim and 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.

Abstract

The generalized numerical index of a Banach space is introduced, and its properties on certain Banach spaces are studied. Ed-dari's theorem on the numerical index is extended to the generalized index and polynomial numerical index of a Banach space. The denseness of numerical strong peak holomorphic functions is also studied.

1. Introduction and Preliminaries

Let and be Banach spaces over a scalar field , where is the real field or the complex field . We denote by and its closed unit ball and unit sphere, respectively. Let be the dual space of . An -homogeneous polynomial from to is a mapping such that there is an -linear (bounded) mapping from to such that for every in . denotes the Banach space of all -homogeneous polynomials from to , endowed with the norm . A mapping is a polynomial if there exist a nonnegative integer and , such that . If , then we say that is a polynomial of degree . We denote by the normed space of all polynomials from to , endowed with the norm . We refer to [1] for background on polynomials on a Banach space.

For two Banach spaces , over a field and a Hausdorff topological space , let Then is a Banach space under the sup norm and is a closed subspace of for each . We just write and instead of and , respectively.

For complex Banach spaces and , we denote that where is the interior of . Then and are closed subspaces of . In case that is the complex scalar field , we write and instead of and , respectively. The closed subspace of consisting of all weakly uniformly continuous functions is denoted by . We denote by one of , , and . Notice that if is finite dimensional, .

Given a real or complex Banach space , we denote by the product topology of the set , where the topologies on and are the norm topology of and the weak- topology of , respectively. The set is a -closed subset of . The spatial numerical range of in is defined [2] by , and the numerical radius of is defined by . Let be an element of . We say that attains its norm if there is some such that . is said to be a (norm) peak function at if there exists a unique such that . It is clear that every (norm) peak function in is norm attaining. A peak function at is said to be a (norm) strong peak function if whenever there is a sequence in with , converges to in . It is easy to see that if is compact, then every peak function is a strong peak function. Given a subspace of , we denote by the set of all points such that there is a strong peak function in with .

Similarly we introduce the notion of numerical peak functions. Let be an element of . If there is some such that , we say [3] that attains its numerical radius. is said ([4, 5]) to be a numerical peak function at if there exist a unique such that . In this case, is said to be the numerical peak point of . It is clear that every numerical peak function in is numerical radius attaining. The numerical peak function at is called a numerical strong peak function if whenever there is a sequence in such that , then converges to in -topology. In this case, is said to be the numerical strong peak point of . We say that a numerical strong peak function at is said to be a very strong numerical peak function if whenever there is a sequence in satisfying , we get and in the norm topology. If is finite dimensional, then every numerical peak function is a very strong numerical peak function.

In 1996, Choi and Kim [6] initiated the study of denseness of norm or numerical radius attaining nonlinear functions, especially homogeneous polynomials on a Banach space. Using the perturbed optimization theorem of Bourgain [7] and Stegall [8], they proved that if a real or complex Banach space has the Radon-Nikodým property, then the set of all norm attaining functions in is norm-dense. For the definition and properties of the Radon-Nikodým property, see [9]. Concerning the numerical radius, it was also shown that if has the Radon-Nikodým property, then the set of all numerical radii attaining functions in is norm-dense. Acosta et al. [10] proved that if a complex Banach space has the Radon-Nikodým property, then the set of all norm attaining functions in is norm-dense. Recently, it was shown in [11] that if has the Radon-Nikodým property, the set of all (norm) strong peak functions in is dense. Concerning the numerical radius, Acosta and Kim [3] showed that the set of all numerical radii attaining functions in is dense if has the Radon-Nikodým property. When is a smooth (complex) Banach space with the Radon-Nikodým property, it is shown in [5] that the set of all numerical strong peak functions is dense in . As a corollary, if and for a measure space , then the set of all norm and numerical strong peak functions in is a dense -subset of . In this case, every numerical strong peak function is a very strong numerical peak function. It is also shown in [5] that the set of all norm and numerical strong peak functions in is a dense -subset of .

Let us briefly sketch the content of this paper. In Section 2, to extend the results of a finite dimensional space to an infinite dimensional space by approximation, we introduce the following notions. A Banach space has the (FPA)-property with if(1)each is a norm-one projection with the finite dimensional range ,(2)given , for every finite-rank operator from into a Banach space and for every finite dimensional subspace of , there is such that As examples, we show that has the (FPA)-property if at least one of the following conditions is satisfied.(a)It has a shrinking and monotone finite-dimensional decomposition.(b), where is a finite measure and .

We show that if has the (FPA)-property, then the set of all polynomials such that there exist a finite dimensional subspace and norm-one projection such that and is a norm, and numerical peak function as a mapping from into is dense in .

A subset of is called a numerical boundary for a subspace of if for every (see [4, 12]). The projections are said to be parallel to a numerical boundary of if each has the image and A projection is said to be strong if whenever is norm-convergent to for a sequence in , is norm-convergent to .

Recall that a Banach space is said to be locally uniformly convex if , and there is a sequence in satisfying , then . Notice that if is locally uniformly convex, then every norm-one projection is strong. We prove that if a smooth Banach space has the (FPA)-property and the corresponding projections are strong and parallel to , then the set of all norm and numerical strong peak functions in is dense. We also prove that if a Banach space has the (FPA)-property with , the corresponding projections are strong, parallel to , and if each is strong, then the set of all very strong numerical and norm strong peak functions is dense in .

In Section 3, we extend the recent result of Ed-dari [13]. Let be a complex Banach space and a subspace of . We introduce the -numerical index by . When for some , the polynomial numerical index is usually denoted by , which was first introduced and studied by Choi et al. [14]. We refer to [1520] for some recent results about polynomial numerical index. For a norm-one projection with range and for any subspace of , define . We prove that if has the (FPA)-property with and the corresponding projections are parallel to a numerical boundary of a subspace , then . In fact, is a decreasing limit of the right-hand side with respect to the inclusion partial order. If is a real Banach space, we get a similar result (see Theorem 14). As a corollary we also extended Ed-dari’s result to the polynomial numerical indices of . In fact, Kim [17] extended Ed-dari’s result [13, Theorem 2.1] to the polynomial numerical indices of (real or complex) of order as follows: Let and be fixed. Then and the sequence is decreasing.

2. Banach Spaces with the (FPA)-Property and Denseness of Numerical Peak Holomorphic Functions

Following [21, Definition 1.g.1], a Banach space has a finite-dimensional Schauder decomposition (FDD for short) if there is a sequence of finite-dimensional spaces such that every has a unique representation of the form , where for every . In such a case, the projections given by are linear and bounded operators. If, moreover, for every , it is satisfied that , the FDD is called shrinking. The FDD is said to be monotone if for every .

The following proposition is easy to prove and its proof is omitted.

Proposition 1. The following two conditions on a Banach space are equivalent. (1)A Banach space has the (FPA)-property. (2)Given , and , there is a norm-one projection such that has a finite rank, and for each and for each , there exist and such that and .

Example 2. Assume that is a complex Banach space satisfying at least one of the following conditions. (1)It has a shrinking and monotone finite-dimensional decomposition.(2), where is a finite measure and .Then has the (FPA)-property.

Proof. Let be a linear operator from to a finite dimensional space and a finite dimensional subspace of . Given , there is an -net in and can be written as for some and .
(1) Suppose that has a shrinking monotone finite-dimensional decomposition. Then there is such that Then for any , hence . For any , there is such that , then because the decomposition is monotone, So taking , we obtained the desired result.
(2) Suppose that . We may assume that is a probability measure. For each , there is such that and . Then there is a sub--algebra generated by finite disjoint subsets such that
Define a projection as . It is clear that is a norm-one projection. For any , On the other hand, for any , there is such that . So We obtained the desired result. The proof is complete.

We will say that a -linear mapping is of finite-type if it can be written as for some , in and in . We will denote by the space of all -linear mappings from to of finite type. If a polynomial is associated with such a -linear mapping, we will say that it is a finite-type polynomial.

Proposition 3. Suppose that a Banach space has the (FPA)-property with . Then the set of all polynomials such that there exists a projection such that and is a norm and numerical peak function as a mapping from to is dense in .

Proof. We follow the ideas in [10]. The subset of continuous polynomials is always dense in . Given and , it is the limit in of sequence of functions defined by . Then belongs to . Thus the Taylor series expansion of at 0 converges uniformly on for all .
We will also use the fact that if is the Taylor series expansion of at 0, then is weakly uniformly continuous on for all .
Since has the (FPA)-property, has the approximation property (see [22, Lemma 3.1]). Then the subspace of -homogeneous polynomials of finite-type restricted on is dense in the subspace of all -homogeneous polynomials which are weakly uniformly continuous on (see [1, Proposition 2.8]). Thus the subspace of the polynomials of finite-type restricted to the closed unit ball of is dense in .
Assume that is a finite-type polynomial that can be written as a finite sum , where each is an homogeneous finite-type polynomial with degree . Consider the symmetric -linear form associated with the corresponding polynomial . Since is a finite-type polynomial, then given by is a linear finite-rank operator for any .
The direct sum of these operators, that is, the operator given by , for all , is also of finite rank.
By the assumption on , given any , there is a norm-one projection with a finite-dimensional range such that and , where is the span of .
Let be the symmetric -linear mapping given by , and let be the associated polynomial. It happens that . Now for , we have Then and Let and . Then . By [5, Theorem 2.9], there is a numerical and norm peak polynomial of degree such that . Setting , . The proof is done.

Remark 4. If is a Banach space satisfying the (FPA)-property, then the set of polynomials in which has a nontrivial invariant subspace and has a fixed point is dense in .

Notice that if is locally uniformly convex, then every norm-one projection is strong. Indeed, suppose that if is a norm-one projection and if in converges to , then shows that and since is locally uniformly convex.

The following lemma is proved in [5].

Lemma 5 (see [5]). Let be a complex Banach space and . Suppose that there are and such that . Then . In particular, .

Theorem 6. Suppose that a smooth Banach space has the (FPA)-property with and the corresponding projections are strong and parallel to . Then the set of all numerical and norm strong peak functions in is dense.

Proof. By Proposition 3, the set of all polynomials such that there exists norm-one projection such that and is a norm and numerical peak function as a mapping from to is dense in .
Fix corresponding and and assume that and for some and , where is the numerical radius of the map .
Suppose that there is a sequence in such that . Then We may assume that the sequence converges to in the norm topology. So . Since is parallel to , . By Lemma 5, So . Since is a numerical peak function, and and .
Since is strong, . Let be the weak- limit point of the sequence . Then and , and implies that since is a numerical strong peak function. Hence is unique because is smooth. Therefore converges weak- to . The proof is complete.

Theorem 7. Suppose that a Banach space space has the (FPA)-property with and the corresponding projections are strong and parallel to . One also assumes that each is strong. Then the set of all very strong numerical and norm strong peak functions is dense in .

Proof. By Proposition 3, the set of all polynomials such that there exists norm-one projection such that and is a norm and numerical peak function as a mapping from to is dense in .
Fix corresponding and and assume that and for some and , where is the numerical radius of the map .
Suppose that there is a sequence in such that . Then We may assume that the sequence converges to in the norm topology. So . Since is parallel to , . By Lemma 5, So . Since is a numerical peak function, and and .
Since is strong, . Fix to be a Hahn-Banach extension of . Let be the weak- limit point of the sequence . Then and and implies that since is a numerical strong peak function so .
Hence and . Now we get by the assumption. This shows that . Therefore and is a very strong numerical peak function at . This completes the proof.

Corollary 8. Suppose that with . Then the set of all very strong numerical and norm strong peak functions is dense in .

Proof. Let be a projection consisting of th natural projections. Then these projections satisfy the conditions in Theorem 7. The proof is done.

3. Generalized Numerical Index

Proposition 9. Let be a (real or complex) Banach spaces and let be a closed subspace of . If has the (FPA)-property with , then . In particular, for each .

Proof. Let . Given , there is a norm one projection with a finite dimensional range such that . Let as a map in and Then there is such that since is finite dimensional. Notice that and so Hence . Therefore .

Proposition 10. Let be a complex Banach space and let be a subspace of with a numerical boundary . Suppose that a norm-one finite dimensional projection is parallel to . Then for any , where is a numerical radius as a function .

Proof. It is clear that . For the converse, choose a sequence in such that Since is in the finite dimensional space , we may assume that converges to and converges to . Then . Thus by Lemma 5, The proof is complete.

For the real Banach spaces, we get the following lemma for a homogeneous polynomial.

Lemma 11. Let be a real or complex Banach space, and let be a -homogeneous polynomial. If there are and such that , then .

Proof. If , then it is clear. So we may assume that . We may assume that . The This completes the proof.

If we use Lemma 11 instead of Lemma 5 in the proof of Proposition 10, we get the following.

Proposition 12. Let be a real or complex Banach space, and let be a numerical boundary of , where is a natural number. Suppose that a norm-one finite dimensional projection is parallel to . Then for any , where is a numerical radius as a function .

Now we get the extensions of the results of Ed-dari [13] and Kim [17] in the complex case.

Theorem 13. Let be a complex Banach space, and let be a subspace of with a numerical boundary . Suppose that the Banach space has the (FPA)-property with and that the corresponding projections are parallel to . Then In fact, is a decreasing limit of the right-hand side with respect to the inclusion partial order.

Proof. For any , by Proposition 10. . Hence and it is easy to see that if , then . Hence . The converse is clear by Proposition 9.

For the general case we get a similar result about the polynomial numerical index if we use Proposition 12 in the proof of Theorem 13.

Theorem 14. Let be a real or complex Banach space, and let be a numerical boundary of , where is a natural number. Suppose that has the (FPA)-property with and that the corresponding projections are parallel to . Then In fact, is a decreasing limit of the right-hand side with respect to the inclusion partial order.

Proposition 15. Let be a real Banach space, and let be a numerical boundary of , where is a natural number. Suppose that has the (FPA)-property with and that the corresponding projections are parallel to . If and , then is one-dimensional.

Proof. We will use the fact [20] that if is a real finite-dimensional Banach space with and , then is one-dimensional. By Theorem 14, we get and for all and ’s are one-dimensional. Suppose on the contrary that is not one-dimensional. Then we can choose two dimensional subspace , and there is with . Then there are and such that for all . Because is two-dimensional, there exists with . So , which is a contradiction to . Therefore, is one-dimensional, and the proof is done.

Example 16. Let be a sequence of finite-dimensional Banach spaces, and consider the following spaces. For each , with the norm is a Banach space with the shrinking and monotone finite-dimensional decomposition with the projections
The space with the norm is also a Banach space with the shrinking and monotone finite-dimensional decomposition with the same projections . Then it is easy to check that is parallel to the projections for each and . So we get the following result. For each and each (or ), where and for complex Banach spaces; we get

Corollary 17. Let be a natural number and . Then for a real or complex case, For the complex case we get

Proof. We give only the first part, since the proof of the next is similar. Let . Then has the (FPA)-property with projections , where each is the th natural projection. Notice that given projections are parallel to . Hence by Theorem 13. Notice that is isometrically isomorphic to .
On the other hand, if we let . Then has (FPA)-property with projections , where each is the conditional expectation with respect to the sub--algebra generated by finitely many disjoint subsets. Hence . Notice also that is isometrically isomorphic to for some . So is isometrically isomorphic to . The proof is complete.

Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009854). The second author is the corresponding author and he was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A1006869).

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