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 X and Y be Banach spaces over a scalar field 𝔽, where 𝔽 is the real field ℝ or the complex field ℂ. We denote by B_X and S_X its closed unit ball and unit sphere, respectively. Let X^* be the dual space of X. An N-homogeneous polynomial P from X to Y is a mapping such that there is an N-linear (bounded) mapping L from X to Y such that P(x) = L(x, …, x) for every x in X. 𝒫(^NX : Y) denotes the Banach space of all N-homogeneous polynomials from X to Y, endowed with the norm . A mapping Q : X → Y is a polynomial if there exist a nonnegative integer m and P_k ∈ 𝒫(^kX : Y), k = 0,1, …, m such that Q = P₀ + P₁ + ⋯+P_m. If P_m ≠ 0, then we say that Q is a polynomial of degree m. We denote by 𝒫(X : Y) the normed space of all polynomials from X to Y, endowed with the norm . We refer to [1] for background on polynomials on a Banach space.

For two Banach spaces X, Y over a field 𝔽 and a Hausdorff topological space K, let

()

Then C_b(K : Y) is a Banach space under the sup norm ∥f∥ : = sup {∥f(t)∥_Y : t ∈ K} and 𝒫(^NX : Y) is a closed subspace of C_b(B_X : Y) for each N ≥ 1. We just write C_b(K) and 𝒫(^NX) instead of C_b(K : 𝔽) and 𝒫(^NX : 𝔽), respectively.

For complex Banach spaces X and Y, we denote that

()

where

is the interior of B_X. Then A_b(B_X : Y) and A_u(B_X : Y) are closed subspaces of C_b(B_X : Y). In case that Y is the complex scalar field ℂ, we write A_b(B_X) and A_u(B_X) instead of A_b(B_X : Y) and A_u(B_X : Y), respectively. The closed subspace of A_u(B_X : Y) consisting of all weakly uniformly continuous functions is denoted by A_wu(B_X : Y). We denote by A(B_X : X) one of A_b(B_X : X), A_u(B_X : X), and A_wu(B_X : X). Notice that if X is finite dimensional, A_b(B_X : X) = A_u(B_X : X) = A_wu(B_X : X).

Given a real or complex Banach space X, we denote by τ the product topology of the set , where the topologies on S_X and are the norm topology of X and the weak-* topology of X^*, respectively. The set is a τ-closed subset of . The spatial numerical range of f in C_b(B_X : X) is defined [2] by W(f) = {x^*(f(x)):(x, x^*) ∈ Π(X)}, and the numerical radius of f is defined by v(f) = sup {|λ | : λ ∈ W(f)}. Let f be an element of C_b(K : X). We say that f attains its norm if there is some t ∈ K such that ∥f∥ = ∥f(t)∥_X. f is said to be a (norm) peak function at t if there exists a unique t ∈ K such that ∥f∥ = ∥f(t)∥_X. It is clear that every (norm) peak function in C_b(K : X) is norm attaining. A peak function f at t is said to be a (norm) strong peak function if whenever there is a sequence in K with , converges to t in K. It is easy to see that if K is compact, then every peak function is a strong peak function. Given a subspace H of C_b(K), we denote by ρH the set of all points t ∈ K such that there is a strong peak function f in H with ∥f∥ = |f(t)|.

Similarly we introduce the notion of numerical peak functions. Let f be an element of C_b(B_X : X). If there is some (x, x^*) ∈ Π(X) such that v(f) = |x^*(f(x))|, we say [3] that f attains its numerical radius. f is said ([4, 5]) to be a numerical peak function at (x, x^*) if there exist a unique (x, x^*) ∈ Π(X) such that v(f) = |x^*(f(x))|. In this case, (x, x^*) is said to be the numerical peak point of f. It is clear that every numerical peak function in C_b(B_X : X) is numerical radius attaining. The numerical peak function f at (x, x^*) is called a numerical strong peak function if whenever there is a sequence in Π(X) such that , then converges to (x, x^*) in τ-topology. In this case, (x, x^*) is said to be the numerical strong peak point of f. We say that a numerical strong peak function f at (x, x^*) is said to be a very strong numerical peak function if whenever there is a sequence in Π(X) satisfying , we get lim _kx_k = x and in the norm topology. If X 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 X has the Radon-Nikodým property, then the set of all norm attaining functions in 𝒫(^kX) 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 X has the Radon-Nikodým property, then the set of all numerical radii attaining functions in 𝒫(^kX : X) is norm-dense. Acosta et al. [10] proved that if a complex Banach space X has the Radon-Nikodým property, then the set of all norm attaining functions in A_b(B_X) is norm-dense. Recently, it was shown in [11] that if X has the Radon-Nikodým property, the set of all (norm) strong peak functions in A_b(B_X) is dense. Concerning the numerical radius, Acosta and Kim [3] showed that the set of all numerical radii attaining functions in A_b(B_X : X) is dense if X has the Radon-Nikodým property. When X 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 A(B_X : X). As a corollary, if 1 < p < ∞ and X = L_p(μ) for a measure space μ, then the set of all norm and numerical strong peak functions in A(B_X : X) is a dense G_δ-subset of A(B_X : X). 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 G_δ-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 X has the (FPA)-property with {π_i, F_i} _i∈I if

(1)
each π_i is a norm-one projection with the finite dimensional range F_i,
(2)
given ϵ > 0, for every finite-rank operator T from X into a Banach space F and for every finite dimensional subspace G of X, there is π_i such that
()

As examples, we show that X 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)
X = L_p(μ), where μ is a finite measure and 1 ≤ p < ∞.

We show that if X has the (FPA)-property, then the set of all polynomials Q ∈ 𝒫(X : X) such that there exist a finite dimensional subspace F and norm-one projection π : X → F such that π∘Q∘π = Q and Q|_F is a norm, and numerical peak function as a mapping from B_F into F is dense in A_wu(B_X : X).

A subset Γ of Π(X) is called a numerical boundary for a subspace H of C_b(B_X : X) if v(f) = sup {|x^*(f(x))|:(x, x^*) ∈ Γ} for every f ∈ H (see [4, 12]). The projections {π_i, F_i} _i∈I are said to be parallel to a numerical boundary Γ of H if each π_i has the image F_i and

()

A projection π : X → X is said to be strong if whenever

is norm-convergent to y ∈ S_X for a sequence

in B_X,

is norm-convergent to y.

Recall that a Banach space X is said to be locally uniformly convex if x ∈ S_X, and there is a sequence {x_n} in B_X satisfying lim _n∥x_n + x∥ = 2, then lim _n∥x_n − x∥ = 0. Notice that if X is locally uniformly convex, then every norm-one projection is strong. We prove that if a smooth Banach space X has the (FPA)-property and the corresponding projections are strong and parallel to Π(X), then the set of all norm and numerical strong peak functions in A_wu(B_X : X) is dense. We also prove that if a Banach space X has the (FPA)-property with {(π_i, F_i)} _i∈I, the corresponding projections are strong, parallel to Π(X), and if each is strong, then the set of all very strong numerical and norm strong peak functions is dense in A_wu(B_X : X).

In Section 3, we extend the recent result of Ed-dari [13]. Let X be a complex Banach space and H a subspace of A_b(B_X : X). We introduce the H-numerical index by N(H): = inf {v(f) : f ∈ H, ∥f∥ = 1}. When H = 𝒫( ^kX : X) for some k ≥ 1, the polynomial numerical index N(H) is usually denoted by n^(k)(X), which was first introduced and studied by Choi et al. [14]. We refer to [15–20] for some recent results about polynomial numerical index. For a norm-one projection π with range F and for any subspace H of A_b(B_X : X), define H_F = {π∘f∘π|_F : B_F → F : f ∈ H}. We prove that if X has the (FPA)-property with {(π_i, F_i)} _i∈I and the corresponding projections are parallel to a numerical boundary of a subspace H, then . In fact, N(H) is a decreasing limit of the right-hand side with respect to the inclusion partial order. If X 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 l_p. In fact, Kim [17] extended Ed-dari’s result [13, Theorem 2.1] to the polynomial numerical indices of (real or complex) l_p of order k as follows: Let 1 < p < ∞ and k ∈ ℕ 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 X has a finite-dimensional Schauder decomposition (FDD for short) if there is a sequence {X_n} of finite-dimensional spaces such that every x ∈ X has a unique representation of the form , where x_n ∈ X_n for every n. In such a case, the projections given by are linear and bounded operators. If, moreover, for every x^* ∈ X^*, it is satisfied that , the FDD is called shrinking. The FDD is said to be monotone if ∥P_n∥ = 1 for every n.

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 X has the (FPA)-property.
(2)
Given ϵ > 0, {x₁, …, x_m} ⊂ X and , there is a norm-one projection P : X → X such that P has a finite rank, and for each i = 1, …, m and for each j = 1, …, n, there exist y_i ∈ X and such that ∥Py_i − x_i∥ ≤ ϵ and .

Example 2. Assume that X is a complex Banach space satisfying at least one of the following conditions.

(1)
It has a shrinking and monotone finite-dimensional decomposition.
(2)
X = L_p(μ), where μ is a finite measure and 1 ≤ p < ∞.

Then X has the (FPA)-property.

Proof. Let T : X → F be a linear operator from X to a finite dimensional space F and G a finite dimensional subspace G of X. Given ϵ > 0, there is an ϵ/3-net {g₁, …, g_n} in B_G and T can be written as for some and y₁, …, y_m ∈ F.

(1) Suppose that X has a shrinking monotone finite-dimensional decomposition. Then there is N ∈ ℕ such that

()

Then for any x ∈ B_X,

()

hence ∥TP_N − T∥ ≤ ϵ. For any x ∈ B_G, there is g_j such that ∥x − g_j∥ ≤ ϵ/3, then because the decomposition is monotone,

()

So taking P = P_N, we obtained the desired result.

(2) Suppose that X = L^p(μ). We may assume that μ is a probability measure. For each 1 ≤ i ≤ m, there is s_i ∈ L_q(μ) such that 1/p + 1/q = 1 and . Then there is a sub-σ-algebra ℱ generated by finite disjoint subsets such that

()

Define a projection P : X → X as Pf = E(f∣ℱ). It is clear that P is a norm-one projection. For any f ∈ B_X,

()

On the other hand, for any f ∈ B_G, there is g_j such that ∥f − g_j∥ ≤ ϵ/3. So

()

We obtained the desired result. The proof is complete.

We will say that a k-linear mapping L : X × ⋯×X → Y is of finite-type if it can be written as

()

for some m ∈ ℕ,

in X^* and y₁, …, y_m in Y. We will denote by L_f(^kX : Y) the space of all k-linear mappings from X to Y of finite type. If a polynomial P is associated with such a k-linear mapping, we will say that it is a finite-type polynomial.

Proposition 3. Suppose that a Banach space X has the (FPA)-property with {(π_i, F_i)} _i. Then the set of all polynomials Q ∈ 𝒫(X : X) such that there exists a projection π_i : X → F_i such that π_i∘Q∘π_i = Q and is a norm and numerical peak function as a mapping from to F_i is dense in A_wu(B_X : X).

Proof. We follow the ideas in [10]. The subset of continuous polynomials is always dense in A_u(B_X : X). Given f ∈ A_u(B_X : X) and n ∈ ℕ, it is the limit in A_u(B_X : X) of sequence of functions {f_n} _n defined by f_n(x): = f((n/(n + 1))x). Then f_n belongs to A_b(((n + 1)/n)B_X : X). Thus the Taylor series expansion of f_n at 0 converges uniformly on B_X for all n.

We will also use the fact that if is the Taylor series expansion of f ∈ A_wu(B_X : X) at 0, then P_k is weakly uniformly continuous on B_X for all k.

Since X has the (FPA)-property, X^* has the approximation property (see [22, Lemma 3.1]). Then the subspace of k-homogeneous polynomials of finite-type restricted on B_X is dense in the subspace of all k-homogeneous polynomials which are weakly uniformly continuous on B_X (see [1, Proposition 2.8]). Thus the subspace of the polynomials of finite-type restricted to the closed unit ball of X is dense in A_wu(B_X : X).

Assume that P is a finite-type polynomial that can be written as a finite sum , where each P_k is an homogeneous finite-type polynomial with degree k. Consider the symmetric k-linear form A_k associated with the corresponding polynomial P_k. Since P_k is a finite-type polynomial, then T_k : X → L_f(^k−1X : X) given by

()

is a linear finite-rank operator for any 1 ≤ k ≤ n.

The direct sum of these operators, that is, the operator

()

given by T(x): = (T₁(x), …, T_n(x)), for all x ∈ X, is also of finite rank.

By the assumption on X, given any ϵ > 0, there is a norm-one projection π : = π_i : X → X with a finite-dimensional range such that ∥T − Tπ∥ ≤ ϵ and ∥π|_G − I_G∥ ≤ ϵ, where G is the span of .

Let B_k be the symmetric k-linear mapping given by B_k : = A_k∘(π, …, π), and let Q_k be the associated polynomial. It happens that Q_k = P_k∘π. Now for ∥x∥ ≤ 1, we have

()

Then ∥P_k∘π − P_k∥ ≤ 4^kϵ and

()

Let R_k = π∘P_k∘π and

. Then ∥R − P∥ ≤ 2n4ⁿϵ. By [5, Theorem 2.9], there is a numerical and norm peak polynomial Q^′ : π(X) → π(X) of degree ≤n such that ∥R|_π(X) − Q^′∥ ≤ ϵ. Setting Q : = Q^′∘π, ∥P − Q∥ ≤ (2n4ⁿ + 2)ϵ. The proof is done.

Remark 4. If X 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 X is locally uniformly convex, then every norm-one projection is strong. Indeed, suppose that if π : X → F is a norm-one projection and if

in B_X converges to y ∈ S_F, then

()

shows that lim _k∥x_k + y∥ = 2 and lim _k∥x_k − y∥ = 0 since X is locally uniformly convex.

The following lemma is proved in [5].

Lemma 5 (see [5].)Let X be a complex Banach space and f ∈ A_b(B_X : X). Suppose that there are y ∈ B_X and such that |y^*(y)| = ∥y^*∥ · ∥y∥. Then |y^*(f(y))| ≤ v(f). In particular, ∥f(0)∥ ≤ v(f).

Theorem 6. Suppose that a smooth Banach space X has the (FPA)-property with {π_i, F_i} _i∈I and the corresponding projections are strong and parallel to Π(X). Then the set of all numerical and norm strong peak functions in A_wu(B_X : X) is dense.

Proof. By Proposition 3, the set of all polynomials Q such that there exists norm-one projection π : = π_i : X → F such that π∘Q∘π = Q and Q|_F is a norm and numerical peak function as a mapping from B_F to F is dense in A_wu(B_X : X).

Fix corresponding Q and π and assume that and ∥Q(y₁)∥ = ∥Q∥ for some and y₁ ∈ B_F, where v_F(Q) is the numerical radius of the map Q|_F : B_F → F.

Suppose that there is a sequence in Π(X) such that . Then

()

We may assume that the sequence

converges to

in the norm topology. So v(Q) = |y^*(Q(y))| ≥ v_F(Q). Since π is parallel to Π(X), |〈y^*, y〉| = ∥y^*∥ · ∥y∥. By Lemma 5,

()

So v(Q) = |y^*(Q(y))| = v_F(Q). Since Q|_F is a numerical peak function, ∥y∥ = 1 = ∥y^*∥ and y = y₀ and

Since π is strong, lim _nx_n = y₀. Let x^* be the weak-* limit point of the sequence . Then x^*(y) = 1 and ∥x^*∥ = 1 = ∥x^*|_F∥, and

()

implies that x^*|_F = y^* since Q|_F is a numerical strong peak function. Hence x^* is unique because X is smooth. Therefore

converges weak-* to x^*. The proof is complete.

Theorem 7. Suppose that a Banach space X space has the (FPA)-property with {π_i, F_i} _i∈I and the corresponding projections are strong and parallel to Π(X). One also assumes that each is strong. Then the set of all very strong numerical and norm strong peak functions is dense in A_wu(B_X : X).

Fix corresponding Q and π and assume that and ∥Q(y₁)∥ = ∥Q∥ for some and y₁ ∈ B_F, where v_F(Q) is the numerical radius of the map Q|_F : B_F → F.

Suppose that there is a sequence in Π(X) such that . Then

()

We may assume that the sequence

converges to

in the norm topology. So v(Q) = |y^*(Q(y))| ≥ v_F(Q). Since π is parallel to Π(X), |〈y^*, y〉| = ∥y^*∥ · ∥y∥. By Lemma 5,

()

So v(Q) = |y^*(Q(y))| = v_F(Q). Since Q|_F is a numerical peak function, ∥y∥ = 1 = ∥y^*∥ and y = y₀ and

Since π is strong, lim _nx_n = y₀. Fix to be a Hahn-Banach extension of y^*. Let x^* be the weak-* limit point of the sequence . Then x^*(y) = 1 and ∥x^*∥ = 1 = ∥π^*(x^*)∥ and

()

implies that π^*(x^*)|_F = y^* since Q|_F is a numerical strong peak function so π^*(x^*) = π^*(x^*).

Hence and ∥π^*(z^*)∥ = 1. Now we get by the assumption. This shows that . Therefore x^* = π^*(z^*) and Q is a very strong numerical peak function at (y, π^*(z^*)). This completes the proof.

Corollary 8. Suppose that X = ℓ_p with 1 < p < ∞. Then the set of all very strong numerical and norm strong peak functions is dense in A_wu(B_X : X).

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

3. Generalized Numerical Index

Proposition 9. Let X be a (real or complex) Banach spaces and let H be a closed subspace of C_b(B_X : X). If X has the (FPA)-property with {π_i, F_i} _i∈I, then . In particular, n^(k)(X) ≥ inf _i∈In^(k)(F_i) for each k ≥ 1.

Proof. Let f ∈ S_H. Given ϵ > 0, there is a norm one projection π with a finite dimensional range F such that ∥π∘f∘π∥ ≥ 1 − ϵ. Let g = π∘f∘π|_F as a map in H_F and

()

Then there is (y, y^*) ∈ Π(H_F) such that

since F is finite dimensional. Notice that (y, π^*(y^*)) ∈ Π(X) and so

()

Hence

. Therefore

Proposition 10. Let X be a complex Banach space and let H be a subspace of A_b(B_X : X) with a numerical boundary Γ. Suppose that a norm-one finite dimensional projection (π, F) is parallel to Γ. Then for any f ∈ H_F,

()

where v_X(f∘π) is a numerical radius as a function f∘π : B_X → X.

Proof. It is clear that v_F(f) ≤ v_X(f∘π). For the converse, choose a sequence in Γ such that

()

Since

is in the finite dimensional space F, we may assume that

converges to y ∈ B_F and

converges to

. Then |〈y^*, y〉 | = ∥y^*∥ · ∥y∥. 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 X be a real or complex Banach space, and let f be a k-homogeneous polynomial. If there are y ∈ B_X and such that |y^*(y)| = ∥y^*∥ · ∥y∥, then |y^*(f(y))| ≤ v(f).

Proof. If y^* = 0, then it is clear. So we may assume that y^* ≠ 0. We may assume that y ≠ 0. 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 X be a real or complex Banach space, and let Γ be a numerical boundary of 𝒫(^kX : X), where k is a natural number. Suppose that a norm-one finite dimensional projection (π, F) is parallel to Γ. Then for any f ∈ 𝒫(^kF : F),

()

where v_X(f∘π) is a numerical radius as a function f∘π : B_X → X.

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

Theorem 13. Let X be a complex Banach space, and let H be a subspace of A_b(B_X : X) with a numerical boundary Γ. Suppose that the Banach space X has the (FPA)-property with {π_i, F_i} _i∈I and that the corresponding projections are parallel to Γ. Then

()

In fact, N(H) is a decreasing limit of the right-hand side with respect to the inclusion partial order.

Proof. For any f ∈ H_F, by Proposition 10. . Hence and it is easy to see that if F_i ⊂ F_j, 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 X be a real or complex Banach space, and let Γ be a numerical boundary of 𝒫(^kX : X), where k is a natural number. Suppose that X has the (FPA)-property with {π_i, F_i} _i∈I and that the corresponding projections are parallel to Γ. Then

()

In fact, n^(k)(X) is a decreasing limit of the right-hand side with respect to the inclusion partial order.

Proposition 15. Let X be a real Banach space, and let Γ be a numerical boundary of 𝒫(^kX : X), where k is a natural number. Suppose that X has the (FPA)-property with {π_i, F_i} _i∈I and that the corresponding projections are parallel to Γ. If n^(k)(X) = 1 and k ≥ 2, then X is one-dimensional.

Proof. We will use the fact [20] that if X is a real finite-dimensional Banach space with n^(k)(X) = 1 and k ≥ 2, then X is one-dimensional. By Theorem 14, we get 1 = n^(k)(X) = inf _i∈In^(k)(F_i) and n^(k)(F_i) = 1 for all i ∈ I and F_i’s are one-dimensional. Suppose on the contrary that X is not one-dimensional. Then we can choose two dimensional subspace G, and there is i ∈ I with ∥id_G − π_i|_G∥ ≤ 1/2. Then there are and a ∈ X such that π_i(x) = x^*(x)a for all x ∈ X. Because G is two-dimensional, there exists w ∈ G∩S_X with x^*(w) = 0. So ∥w − π_i(w)∥ = ∥w∥ ≤ 1/2, which is a contradiction to ∥w∥ = 1. Therefore, X 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 1 < p < ∞,

()

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 P_n. Then it is easy to check that Π(X_p) is parallel to the projections (P_n, P_n(X_p)) for each 1 < p < ∞ and p = 0. So we get the following result. For each k ∈ ℕ and each 1 < p < ∞ (or p = 0),

()

where

and for complex Banach spaces; we get

()

Corollary 17. Let k ≥ 1 be a natural number and 1 < p < ∞. 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 H = 𝒫( ^kℓ_p). Then ℓ_p has the (FPA)-property with projections , where each π_i is the ith natural projection. Notice that given projections are parallel to Π(X). Hence by Theorem 13. Notice that is isometrically isomorphic to .

On the other hand, if we let H = 𝒫( ^kL_p(0,1)). Then L_p(0,1) has (FPA)-property with projections {π_i, F_i}, where each π_i is the conditional expectation with respect to the sub-σ-algebra generated by finitely many disjoint subsets. Hence . Notice also that F_i is isometrically isomorphic to for some m. 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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Generalized Numerical Index and Denseness of Numerical Peak Holomorphic Functions on a Banach Space

Abstract

1. Introduction and Preliminaries

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

3. Generalized Numerical Index

Acknowledgments

References

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

Abstract

1. Introduction and Preliminaries

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

3. Generalized Numerical Index

Acknowledgments

References

References

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