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

and Applied Analysis 3 A b (B X : X). We introduce the H-numerical index by N(H) := inf{V(f) : f ∈ H, ‖f‖ = 1}. WhenH = P(X : X) for some k ≥ 1, the polynomial numerical index N(H) is usually denoted by n(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 normone 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 N(H) = inf i∈I N(H Fi ). 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 (seeTheorem 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 n(l p ) = inf{n(l p ) : m ∈ N} and the sequence

radius of  is defined by V() = sup{|| :  ∈ ()}.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 {  } ∞ =1 in  with lim  ‖(  )‖  = ‖‖, {  } ∞ =1 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 ‖‖ = |()|.
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 P(  ) 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 P(   : ) is normdense.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 1 <  < ∞ 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 (  1 :  1 ) is a dense   -subset of (  1 :  1 ).
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  > 0, 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.
We show that if  has the (FPA)-property, then the set of all polynomials  ∈ P( : ) 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 projection  :  →  is said to be strong if whenever Recall that a Banach space  is said to be locally uniformly convex if  ∈   , and there is a sequence {  } in   satisfying lim  ‖  + ‖ = 2, then lim  ‖  − ‖ = 0. 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 Π(), 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 Π(), and if each  *  :  * →  * is strong, then the set of all very strong numerical and norm strong peak functions is dense in   (  : ).

Banach Spaces with the (FPA)-Property and Denseness of Numerical Peak Holomorphic Functions
Following [21, Definition 1.g.1], a Banach space  has a finitedimensional Schauder decomposition (FDD for short) if there is a sequence {  } of finite-dimensional spaces such that every  ∈  has a unique representation of the form  = ∑ ∞ =1   , where   ∈   for every .In such a case, the projections given by   () = ∑  =1   are linear and bounded operators.If, moreover, for every  * ∈  * , it is satisfied that ‖ *   * −  * ‖ → 0, the FDD is called shrinking.The FDD is said to be monotone if ‖  ‖ = 1 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) It has a shrinking and monotone finite-dimensional decomposition.
Then  has the (FPA)-property.
Proof.Let  :  →  be a linear operator from  to a finite dimensional space  and  a finite dimensional subspace  of .Given  > 0, there is an /3-net { So taking  =   , we obtained the desired result.
(2) Suppose that  =   ().We may assume that  is a probability measure.For each 1 ≤  ≤ , there is   ∈   () such that 1/ + 1/ = 1 and 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  ∈ N,  * 1,1 , . . .,  * , in  * and  1 , . . .,   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 finitetype polynomial.Proposition 3. Suppose that a Banach space  has the (FPA)property with {(  ,   )}  .Then the set of all polynomials  ∈ P( : ) such that there exists a projection   :  →   such that   ∘  ∘   =  and |   is a norm and numerical peak function as a mapping from    to   is dense in   (  : ).
We will also use the fact that if ∑ ∞ =0   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  = ∑  =0   , where each   is an homogeneous finite-type polynomial with degree .
The following lemma is proved in [5].
Theorem 7. Suppose that a Banach space  space has the (FPA)-property with {  ,   } ∈ and the corresponding projections are strong and parallel to Π().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   (  : ).
Suppose that there is a sequence We may assume that the sequence Proof.Let {  ,   } ∞ =1 be a projection consisting of th natural projections.Then these projections satisfy the conditions in Theorem 7. The proof is done.
Proof.Let  ∈   .Given  > 0, there is a norm one projection  with a finite dimensional range  such that ‖∘∘‖ ≥ 1−.Let  =  ∘  ∘ |  as a map in   and Then there is (, 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 | * (())| ≤ V().
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 Γ be a numerical boundary of P(   : ), where  is a natural number.Suppose that a norm-one finite dimensional projection (, ) is parallel to Γ. Then for any  ∈ P(   : ), where V  ( ∘ ) 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 Γ.Suppose that the Banach space  has the (FPA)-property with {  ,   } ∈ and that the corresponding projections are parallel to Γ. Then  () = inf ∈  (   ) . (30) In fact, () is a decreasing limit of the right-hand side with respect to the inclusion partial order.
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 Γ be a numerical boundary of P(   : ), where  is a natural number.Suppose that  has the (FPA)-property with {  ,   } ∈ and that the corresponding projections are parallel to Γ. Then  () () = inf ∈  () (  ) . (31) 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 Γ be a numerical boundary of P(   : ), where  is a natural number.Suppose that  has the (FPA)-property with {  ,   } ∈ and that the corresponding projections are parallel to Γ.If  () () = 1 and  ≥ 2, then  is one-dimensional.
Theorem 6. Suppose that a smooth Banach space  has the (FPA)-property with {  ,   } ∈ and the corresponding projections are strong and parallel to Π().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 V  () = | * 0 (( 0 ))| and ‖( 1 )‖ = ‖‖ for some ( * 0 ,  0 ) ∈ Π()      ,  ( (  ))⟩     → V () .