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

Authors: Sung Guen Kim; Han Ju Lee
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction and Preliminaries

LetX 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 BX and SX its closed unit ball and unit sphere, respectively. Let X* be the dual space of X. An N-homogeneous polynomialP 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 ∥P∥=supx∈BX∥P(x)∥. A mapping Q:X→Y is a polynomial if there exist a nonnegative integer m and Pk∈𝒫(kX:Y), k=0,1,…,m such that Q=P0+P1+⋯+Pm. If Pm≠0, then we say that Q is a polynomial of degreem. We denote by 𝒫(X:Y) the normed space of all polynomials from X to Y, endowed with the norm ∥Q∥=supx∈BX∥Q(x)∥. We refer to [1] for background on polynomials on a Banach space.For two Banach spacesX, Y over a field 𝔽 and a Hausdorff topological space K, let
(1)Cb(K:Y):={f:K⟶Y:fbeaboundedcontinuousfunctiononKf:K⟶Y:fisabounded}.
Then Cb(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 Cb(BX:Y) for each N≥1. We just write Cb(K) and 𝒫(NX) instead of Cb(K:𝔽) and 𝒫(NX:𝔽), respectively.For complex Banach spacesX and Y, we denote that
(2)Ab(BX:Y):={f∈Cb(BX:Y):fisholomorphiconBX∘}Au(BX:Y):={f∈Ab(BX:Y):fisuniformlycontinuousf∈Ab(BX:Y):fis},
where BX∘ is the interior of BX. Then Ab(BX:Y) and Au(BX:Y) are closed subspaces of Cb(BX:Y). In case that Y is the complex scalar field ℂ, we write Ab(BX) and Au(BX) instead of Ab(BX:Y) and Au(BX:Y), respectively. The closed subspace of Au(BX:Y) consisting of all weakly uniformly continuous functions is denoted by Awu(BX:Y). We denote by A(BX:X) one of Ab(BX:X), Au(BX:X), and Awu(BX:X). Notice that if X is finite dimensional, Ab(BX:X)=Au(BX:X)=Awu(BX:X).Given a real or complex Banach spaceX, we denote by τ the product topology of the set SX×SX*, where the topologies on SX and SX* are the norm topology of X and the weak-* topology of X*, respectively. The set Π(X):={(x,x*)∈SX×SX*:∥x∥=∥x*∥=1=x*(x)} is a τ-closed subset of SX×SX*. The spatial numerical range of f in Cb(BX: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 Cb(K:X). We say that fattains 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 Cb(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 {tk}k=1∞ in K with limk∥f(tk)∥X=∥f∥, {tk}k=1∞ 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 Cb(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. Letf be an element of Cb(BX:X). If there is some (x,x*)∈Π(X) such that v(f)=|x*(f(x))|, we say [3] that fattains 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 Cb(BX: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 {(xk,xk*)}k=1∞ in Π(X) such that limk|xk*(f(xk))|=v(f), then {(xk,xk*)}k=1∞ 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 {(xk,xk*)}k=1∞ in Π(X) satisfying limn|xk*(f(xk))|=v(f), we get limkxk=x and limkxk*=x* 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 Ab(BX) 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 Ab(BX) is dense. Concerning the numerical radius, Acosta and Kim [3] showed that the set of all numerical radii attaining functions in Ab(BX: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(BX:X). As a corollary, if 1<p<∞ and X=Lp(μ) for a measure space μ, then the set of all norm and numerical strong peak functions in A(BX:X) is a dense Gδ-subset of A(BX: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 A(Bl1:l1) is a dense Gδ-subset of A(Bl1:l1).Let us briefly sketch the content of this paper. In Section2, 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 {πi,Fi}i∈I if(1)
eachπi is a norm-one projection with the finite dimensional range Fi,(2)
givenϵ>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 πi such that
(3)∥T-Tπi∥≤ϵ,∥IG-πi|G∥≤ϵ.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=Lp(μ), where μ is a finite measure and 1≤p<∞.We show that ifX 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 BF into F is dense in Awu(BX:X).A subsetΓ of Π(X) is called a numerical boundary for a subspace H of Cb(BX:X) if v(f)=sup{|x*(f(x))|:(x,x*)∈Γ} for every f∈H (see [4, 12]). The projections {πi,Fi}i∈I are said to be parallel to a numerical boundary Γ of H if each πi has the image Fi and
(4)|〈x*|Fi,πi(x)〉|=∥x*|Fi∥·∥πi(x)∥,∀(x,x*)∈Γ,∀i∈I.
A projection π:X→X is said to be strong if whenever {π(xk)}k=1∞ is norm-convergent to y∈SX for a sequence {xk}k=1∞ in BX, {xk}k=1∞ is norm-convergent to y.Recall that a Banach spaceX is said to be locally uniformly convex if x∈SX, and there is a sequence {xn} in BX satisfying limn∥xn+x∥=2, then limn∥xn-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 Awu(BX:X) is dense. We also prove that if a Banach space X has the (FPA)-property with {(πi,Fi)}i∈I, the corresponding projections are strong, parallel to Π(X), and if each πi*:X*→X* is strong, then the set of all very strong numerical and norm strong peak functions is dense in Awu(BX:X).In Section3, we extend the recent result of Ed-dari [13]. Let X be a complex Banach space and H a subspace of Ab(BX: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 Ab(BX:X), define HF={π∘f∘π|F:BF→F:f∈H}. We prove that if X has the (FPA)-property with {(πi,Fi)}i∈I and the corresponding projections are parallel to a numerical boundary of a subspace H, then N(H)=infi∈IN(HFi). 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 lp. In fact, Kim [17] extended Ed-dari’s result [13, Theorem 2.1] to the polynomial numerical indices of (real or complex) lp of order k as follows: Let 1<p<∞ and k∈ℕ be fixed. Then n(k)(lp)=inf{n(k)(lpm):m∈ℕ} and the sequence {n(k)(lpm)}m∈ℕ 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 {Xn} of finite-dimensional spaces such that every x∈X has a unique representation of the form x=∑n=1∞‍xn, where xn∈Xn for every n. In such a case, the projections given by Pn(x)=∑i=1n‍xi are linear and bounded operators. If, moreover, for every x*∈X*, it is satisfied that ∥Pn*x*-x*∥→0, the FDD is called shrinking. The FDD is said to be monotone if ∥Pn∥=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 spaceXhas the (FPA)-property.(2)
Givenϵ>0,{x1,…,xm}⊂Xand{x1*,…,xn*}⊂X*, there is a norm-one projectionP:X→Xsuch thatPhas a finite rank, and for eachi=1,…,mand for eachj=1,…,n, there existyi∈Xandyj*∈X*such that∥Pyi-xi∥≤ϵand∥P*yj*-xj*∥≤ϵ.Example 2.
Assume thatX 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=Lp(μ), where μ is a finite measure and 1≤p<∞.Then X has the (FPA)-property.Proof.
LetT: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,…,gn} in BG and T can be written as ∑i=1m‍xi*⊗yi for some x1*,…,xm*∈X* and y1,…,ym∈F.
(1) Suppose thatX has a shrinking monotone finite-dimensional decomposition. Then there is N∈ℕ such that
(5)max1≤i≤m∥yi∥·∑i=1m‍∥PN*xi*-xi*∥≤ϵ,max1≤j≤n∥PNgj-gj∥≤ϵ3.
Then for any x∈BX,
(6)∥TPNx-Tx∥=∥∑i=1m‍(PN*xi*)(x)yi-∑i=1m‍xi*(x)yi∥≤max1≤j≤n∥yi∥·∑i=1m‍∥PN*xi*-xi*∥≤ϵ,
hence ∥TPN-T∥≤ϵ. For any x∈BG, there is gj such that ∥x-gj∥≤ϵ/3, then because the decomposition is monotone,
(7)∥PNx-x∥≤∥PN(x-gj)∥+∥PNgj-gj∥+∥x-gj∥≤2∥x-gj∥+∥PNgj-gj∥≤ϵ.
So taking P=PN, we obtained the desired result.
(2) Suppose thatX=Lp(μ). We may assume that μ is a probability measure. For each 1≤i≤m, there is si∈Lq(μ) such that 1/p+1/q=1 and xi*(f)=∫‍fsidμ(f∈Lp(μ)). Then there is a sub-σ-algebra ℱ generated by finite disjoint subsets such that
(8)max1≤j≤n∥yi∥·∑i=1m∥E(si∣ℱ)-si∥q‍≤ϵ2,max1≤i≤n∥E(gi∣ℱ)-gi∥p≤ϵ3.
Define a projectionP:X→X as Pf=E(f∣ℱ). It is clear that P is a norm-one projection. For any f∈BX,
(9)∥TPf-Tf∥=∥∑i=1m‍(xi*)(Pf)yi-∑i=1m‍xi*(f)yi∥≤max1≤j≤n∥yi∥·∑i=1m‍|xi*(Pf)-xi*(f)|≤max1≤j≤n∥yi∥·∑i=1m‍|∫K‍(E(f∣ℱ)-f)E(si∣ℱ)dμ|+max1≤j≤n∥yi∥·∑i=1m‍|∫K‍(E(f∣ℱ)-f)(E(si∣ℱ)-si)dμ|=0+max1≤j≤n∥yi∥·∑i=1m‍|∫K‍(E(f∣ℱ)-f)×(E(si∣ℱ)-si)dμ∫K‍|≤max1≤j≤n∥yi∥·2∑i=1m‍∥f∥p∥E(si∣ℱ)-si∥q≤ϵ.
On the other hand, for any f∈BG, there is gj such that ∥f-gj∥≤ϵ/3. So
(10)∥Pf-f∥≤∥P(f-gj)∥+∥Pgj-gj∥+∥x-gj∥≤2∥f-gj∥+∥Pgj-gj∥≤ϵ.
We obtained the desired result. The proof is complete.We will say that ak-linear mapping L:X×⋯×X→Y is of finite-type if it can be written as
(11)L(x1,…,xk)=∑i=1m‍x1,i*(x1)⋯xk,i*(xk)yi,∀x1,…,xk∈X
for some m∈ℕ, x1,1*,…,xk,m* in X* and y1,…,ym in Y. We will denote by Lf(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 spaceX has the (FPA)-property with {(πi,Fi)}i. Then the set of all polynomials Q∈𝒫(X:X) such that there exists a projection πi:X→Fi such that πi∘Q∘πi=Q and Q|Fi is a norm and numerical peak function as a mapping from BFi to Fi is dense in Awu(BX:X).Proof.
We follow the ideas in [10]. The subset of continuous polynomials is always dense in Au(BX:X). Given f∈Au(BX:X) and n∈ℕ, it is the limit in Au(BX:X) of sequence of functions {fn}n defined by fn(x):=f((n/(n+1))x). Then fn belongs to Ab(((n+1)/n)BX:X). Thus the Taylor series expansion of fn at 0 converges uniformly on BX for all n.
We will also use the fact that if∑k=0∞‍Pk is the Taylor series expansion of f∈Awu(BX:X) at 0, then Pk is weakly uniformly continuous on BX for all k.
SinceX 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 BX is dense in the subspace of all k-homogeneous polynomials which are weakly uniformly continuous on BX (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 Awu(BX:X).
Assume thatP is a finite-type polynomial that can be written as a finite sum P=∑k=0n‍Pk, where each Pk is an homogeneous finite-type polynomial with degree k. Consider the symmetric k-linear form Ak associated with the corresponding polynomial Pk. Since Pk is a finite-type polynomial, then Tk:X→Lf(k-1X:X) given by
(12)Tk(x)(x1,…,xk-1):=Ak(x,x1,…,xk-1),∀x∈X
is a linear finite-rank operator for any 1≤k≤n.
The direct sum of these operators, that is, the operator(13)T:X⟶⨁k=1nLf(k-1X:X)
given by T(x):=(T1(x),…,Tn(x)), for all x∈X, is also of finite rank.
By the assumption onX, given any ϵ>0, there is a norm-one projection π:=πi:X→X with a finite-dimensional range such that ∥T-Tπ∥≤ϵ and ∥π|G-IG∥≤ϵ, where G is the span of ⋃k=1n‍Pk(X).
LetBk be the symmetric k-linear mapping given by Bk:=Ak∘(π,…,π), and let Qk be the associated polynomial. It happens that Qk=Pk∘π. Now for ∥x∥≤1, we have
(14)∥Pk∘π(x)-Pk(x)∥=∥∑j=0k-1‍(kj)Ak((x-π(x))k-j,π(x)j)∥=∥∑j=0k-1‍(kj)(Tk-Tk∘π)(x)((x-π(x))k-j-1,π(x)j)∥≤∑j=0k-1‍(kj)∥Tk-Tk∘π∥∥x∥∥x-π(x)∥k-j-1∥π(x)∥j≤ϵ∑j=0k-1‍(kj)2k-j-1≤4kϵ.
Then ∥Pk∘π-Pk∥≤4kϵ and
(15)∥π∘Pk∘π-Pk∥≤∥π∘Pk∘π-π∘Pk∥+∥π∘Pk-Pk∥≤2·4kϵ.
Let Rk=π∘Pk∘π and R=P0+∑k=1n‍Rk. 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.
IfX is a Banach space satisfying the (FPA)-property, then the set of polynomials in BAwu(BX:X) which has a nontrivial invariant subspace and has a fixed point is dense in BAwu(BX:X).Notice that ifX 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 {π(xk)}k=1∞ in BX converges to y∈SF, then
(16)1=limk∥π(xk)+y2∥=limk∥π(xk+y)2∥≤limk∥xk+y2∥≤1
shows that limk∥xk+y∥=2 and limk∥xk-y∥=0 since X is locally uniformly convex.The following lemma is proved in [5].Lemma 5 (see [5]).
LetX be a complex Banach space and f∈Ab(BX:X). Suppose that there are y∈BX and y*∈BX* 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 spaceX has the (FPA)-property with {πi,Fi}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 Awu(BX:X) is dense.Proof.
By Proposition3, the set of all polynomials Q such that there exists norm-one projection π:=πi:X→F such that π∘Q∘π=Q and Q|F is a norm and numerical peak function as a mapping from BF to F is dense in Awu(BX:X).
Fix correspondingQ and π and assume that vF(Q)=|y0*(Q(y0))| and ∥Q(y1)∥=∥Q∥ for some (y0*,y0)∈Π(F) and y1∈BF, where vF(Q) is the numerical radius of the map Q|F:BF→F.
Suppose that there is a sequence{(xk,xk*)}k=1∞ in Π(X) such that limk|xk*(Q(xk))|=v(Q). Then
(17)|〈xk*,Q(xk)〉|=|〈xk*|F,Q(π(xk))〉|⟶v(Q).
We may assume that the sequence {(π(xk),xk*|F)}k=1∞ converges to (y,y*)∈BF×BF* in the norm topology. So v(Q)=|y*(Q(y))|≥vF(Q). Since π is parallel to Π(X), |〈y*,y〉|=∥y*∥·∥y∥. By Lemma 5,
(18)v(Q)=|y*(Q(y))|≤vF(Q).
So v(Q)=|y*(Q(y))|=vF(Q). Since Q|F is a numerical peak function, ∥y∥=1=∥y*∥ and y=y0 and y*=y0*.
Sinceπ is strong, limnxn=y0. Let x* be the weak-* limit point of the sequence {xn*}. Then x*(y)=1 and ∥x*∥=1=∥x*|F∥, and
(19)v(Q)=|x*(Q(y))|=|y*(Q(y))|=vF(Q)
implies that x*|F=y* since Q|F is a numerical strong peak function. Hence x* is unique because X is smooth. Therefore {xn*}n=1∞ converges weak-* to x*. The proof is complete.Theorem 7.
Suppose that a Banach spaceX space has the (FPA)-property with {πi,Fi}i∈I and the corresponding projections are strong and parallel to Π(X). One also assumes that each πi*:X*→X* is strong. Then the set of all very strong numerical and norm strong peak functions is dense in Awu(BX:X).Proof.
By Proposition3, the set of all polynomials Q such that there exists norm-one projection π:=πi:X→F such that π∘Q∘π=Q and Q|F is a norm and numerical peak function as a mapping from BF to F is dense in Awu(BX:X).
Fix correspondingQ and π and assume that vF(Q)=|y0*(Q(y0))| and ∥Q(y1)∥=∥Q∥ for some (y0*,y0)∈Π(F) and y1∈BF, where vF(Q) is the numerical radius of the map Q|F:BF→F.
Suppose that there is a sequence{(xk,xk*)}k=1∞ in Π(X) such that limk|xk*(Q(xk))|=v(Q). Then
(20)|〈xk*,Q(xk)〉|=|〈xk*|F,Q(π(xk))〉|⟶v(Q).
We may assume that the sequence {(π(xk),xk*|F)}k=1∞ converges to (y,y*)∈BF×BF* in the norm topology. So v(Q)=|y*(Q(y))|≥vF(Q). Since π is parallel to Π(X), |〈y*,y〉|=∥y*∥·∥y∥. By Lemma 5,
(21)v(Q)=|y*(Q(y))|≤vF(Q).
So v(Q)=|y*(Q(y))|=vF(Q). Since Q|F is a numerical peak function, ∥y∥=1=∥y*∥ and y=y0 and y*=y0*.
Sinceπ is strong, limnxn=y0. Fix z*∈SX* to be a Hahn-Banach extension of y*. Let x* be the weak-* limit point of the sequence {xn*}n=1∞. Then x*(y)=1 and ∥x*∥=1=∥π*(x*)∥ and
(22)v(Q)=|x*(Q(y))|=|y*(Q(y))|=vF(Q)
implies that π*(x*)|F=y* since Q|F is a numerical strong peak function so π*(x*)=π*(x*).
Hencelimnπ*(xn*)=π*(z*) and ∥π*(z*)∥=1. Now we get ∥xn*-π*(z*)∥→0 by the assumption. This shows that limn∥xn*-π*(z*)∥=0. Therefore x*=π*(z*) and Q is a very strong numerical peak function at (y,π*(z*)). This completes the proof.Corollary 8.
Suppose thatX=ℓp with 1<p<∞. Then the set of all very strong numerical and norm strong peak functions is dense in Awu(BX:X).Proof.
Let{πi,Fi}i=1∞ 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.
LetX be a (real or complex) Banach spaces and let H be a closed subspace of Cb(BX:X). If X has the (FPA)-property with {πi,Fi}i∈I, then N(H)≥infi∈IN(HFi). In particular, n(k)(X)≥infi∈In(k)(Fi) for each k≥1.Proof.
Letf∈SH. 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 HF and
(23)vF(g)≥N(HF)∥g∥≥N(HF)(1-ϵ).
Then there is (y,y*)∈Π(HF) such that vHF(g)=|y*(g(y))| since F is finite dimensional. Notice that (y,π*(y*))∈Π(X) and so
(24)vF(g)=|π*y*(f(π(y)))|=|π*x*(f(y))|≤vH(f).
Hence vH(f)≥(1-ϵ)N(HF)≥(1-ϵ)infi∈IN(HFi). Therefore N(H)≥infi∈IN(HFi).Proposition 10.
LetX be a complex Banach space and let H be a subspace of Ab(BX:X) with a numerical boundary Γ. Suppose that a norm-one finite dimensional projection (π,F) is parallel to Γ. Then for any f∈HF,
(25)vF(f)=vX(f∘π),
where vX(f∘π) is a numerical radius as a function f∘π:BX→X.Proof.
It is clear thatvF(f)≤vX(f∘π). For the converse, choose a sequence {(xn,xn*)}n=1∞ in Γ such that
(26)vX(f∘π)=limn|xn*(f(π(xn)))|=limn〈xn*|F,f(π(xn))〉.
Since {π(xn)}n=1∞ is in the finite dimensional space F, we may assume that {π(xn)}n=1∞ converges to y∈BF and {xn*|F}n=1∞ converges to y*∈BF*. Then |〈y*,y〉|=∥y*∥·∥y∥. Thus by Lemma 5,
(27)vX(f∘π)=|y*(f(y))|≤vF(f).
The proof is complete.For the real Banach spaces, we get the following lemma for a homogeneous polynomial.Lemma 11.
LetX be a real or complex Banach space, and let f be a k-homogeneous polynomial. If there are y∈BX and y*∈BX* such that |y*(y)|=∥y*∥·∥y∥, then |y*(f(y))|≤v(f).Proof.
Ify*=0, then it is clear. So we may assume that y*≠0. We may assume that y≠0. The
(28)|y*(f(y))|≤|y*∥y*∥∥y∥k(f(y))|=|y*∥y*∥(f(y∥y∥))|≤v(f).
This completes the proof.If we use Lemma11 instead of Lemma 5 in the proof of Proposition 10, we get the following.Proposition 12.
LetX 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),
(29)vF(f)=vX(f∘π),
where vX(f∘π) is a numerical radius as a function f∘π:BX→X.Now we get the extensions of the results of Ed-dari [13] and Kim [17] in the complex case.Theorem 13.
LetX be a complex Banach space, and let H be a subspace of Ab(BX:X) with a numerical boundary Γ. Suppose that the Banach space X has the (FPA)-property with {πi,Fi}i∈I and that the corresponding projections are parallel to Γ. Then
(30)N(H)=infi∈IN(HFi).
In fact, N(H) is a decreasing limit of the right-hand side with respect to the inclusion partial order.Proof.
For anyf∈HF, vFi(f)=vX(f∘πi) by Proposition 10. vFi(f)=vX(f∘π)≥∥f∘π∥N(H)=∥f∥N(H). Hence N(HFi)≥N(H) and it is easy to see that if Fi⊂Fj, then N(H)≤N(HFj)≤N(HFi). Hence N(H)≤infi∈IN(HFi). The converse is clear by Proposition 9.For the general case we get a similar result about the polynomial numerical index if we use Proposition12 in the proof of Theorem 13.Theorem 14.
LetX 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 {πi,Fi}i∈I and that the corresponding projections are parallel to Γ. Then
(31)n(k)(X)=infi∈In(k)(Fi).
In fact, n(k)(X) is a decreasing limit of the right-hand side with respect to the inclusion partial order.Proposition 15.
LetX 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 {πi,Fi}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)=infi∈In(k)(Fi) and n(k)(Fi)=1 for all i∈I and Fi’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 ∥idG-πi|G∥≤1/2. Then there are x*∈SX* and a∈X such that πi(x)=x*(x)a for all x∈X. Because G is two-dimensional, there exists w∈G∩SX with x*(w)=0. So ∥w-πi(w)∥=∥w∥≤1/2, which is a contradiction to ∥w∥=1. Therefore, X is one-dimensional, and the proof is done.Example 16.
Let(Fi)i=1∞ be a sequence of finite-dimensional Banach spaces, and consider the following spaces. For each 1<p<∞,
(32)Xp:={(xi)i=1∞:xi∈Fi,(∥xi∥)i=1∞∈ℓp}
with the norm
(33)∥(xi)i=1∞∥=(∑i=1∞‍∥xi∥p)1/p
is a Banach space with the shrinking and monotone finite-dimensional decomposition with the projections
(34)Pn((xi)i=1∞)=(x1,…,xn,0,0,…),(n∈ℕ).
The spaceX0:={(xi)i=1∞:xi∈Fi,(∥xi∥)i=1∞∈c0} with the norm ∥(xi)i=1∞∥=supi∈ℕ∥xi∥ is also a Banach space with the shrinking and monotone finite-dimensional decomposition with the same projections Pn. Then it is easy to check that Π(Xp) is parallel to the projections (Pn,Pn(Xp)) for each 1<p<∞ and p=0. So we get the following result. For each k∈ℕ and each 1<p<∞ (or p=0),
(35)n(k)(Xp)=limn→∞n(k)(Xpn),
where Xpn=F1⊕pF2⊕p⋯⊕pFn and for complex Banach spaces; we get
(36)N(A(BXp:Xp))=limn→∞N(A(BXpn:Xpn)).Corollary 17.
Letk≥1 be a natural number and 1<p<∞. Then for a real or complex case,
(37)limm→∞n(k)(ℓpm)=n(k)(ℓp)≤n(k)(Lp(0,1)).
For the complex case we get
(38)limm→∞N(Ab(Bℓpm:ℓpm))=N(Ab(Bℓp:ℓp))≤N(Ab(BLp(0,1):Lp(0,1))).Proof.
We give only the first part, since the proof of the next is similar. LetH=𝒫(kℓp). Then ℓp has the (FPA)-property with projections {πi,ℓpi}i=1∞, where each πi is the ith natural projection. Notice that given projections are parallel to Π(X). Hence N(H)=infi∈IN(HFi) by Theorem 13. Notice that HFi is isometrically isomorphic to 𝒫(kℓpi).
On the other hand, if we letH=𝒫(kLp(0,1)). Then Lp(0,1) has (FPA)-property with projections {πi,Fi}, where each πi is the conditional expectation with respect to the sub-σ-algebra generated by finitely many disjoint subsets. Hence N(H)≥infi∈IN(HFi). Notice also that Fi is isometrically isomorphic to ℓpm for some m. So HFi is isometrically isomorphic to 𝒫(kℓpm). The proof is complete.



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*Source: 380475-2013-09-26.xml*