Polynomial Numerical Index Research Articles

SOME OBSERVATIONS ON THE NUMERICAL INDEX AND THE POLYNOMIAL NUMERICAL INDEX

In this paper, we study both the numerical index and the polynomial numerical index. First, we give a sufficient condition for a Banach space X to have lushness. Second, we study the relation between the renormings of a Banach space and the k-order polynomial numerical index. This shows that every real Banach spaces of dimension greater that 1 can be renormed to have 2-order polynomial numerical index for any .

Three kinds of numerical indices of a Banach space II

For a Banach space E and a positive integer k, we study about three kinds of numerical indices of E, the multilinear numerical index (E), the symmetric multilinear numerical index (E) and the polynomial numerical index (E). First we show that (E**) ≤ (E) for I=m, s and present some inequalities among (E), (E) and (E). We also prove that if E is a strictly convex Banach space, then (E)=0 for every k ≥ 2.

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

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.

Polynomial numerical indices of Banach spaces with 1-unconditional bases

The only infinite-dimensional complex space with 1-unconditional basis which has polynomial numerical index of order 2 equal to 1 is c0. In the real case, there is no space of this type. We also show that, in the complex case, if X is an infinite-dimensional Banach sequence space with absolute norm whose Köthe dual is norming and has polynomial numerical index of order 2 equal to 1, then c0⊂X⊂ℓ∞. In the real case, again there is no space of this type.

Polynomial numerical indices of Banach spaces with absolute norm

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon–Nikodým property or is Asplund, satisfies n ( 2 ) ( X ) < 1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon–Nikodým property which satisfy n ( 2 ) ( X ) = 1 are the spaces ℓ ∞ m . Also, the only Asplund complex space X with absolute norm which satisfies n ( 2 ) ( X ) = 1 is c 0 ( Λ ) .

Two-dimensional Banach spaces with polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of C b ( K ) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of P ( X n ) , then the set of all strongly norm attaining elements in P ( X n ) is dense. In particular, the set of all points at which the norm of P ( X n ) is Fréchet differentiable is a dense G δ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to ℓ ∞ n . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.

Banach spaces with polynomial numerical index 1

We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\'ym property.

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study in this paper the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K,X), and L∞(μ,X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every σ-finite measure μ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k > 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X∗∗ is the least possible, namely k k 1−k . We also give new examples of Banach spaces with the polynomial Daugavet property, namely L∞(μ,X) when μ is atomless, and Cw(K,X), Cw∗ (K,X ∗) when K is perfect.

On the polynomial numerical index of the real spaces [formula omitted], [formula omitted] and [formula omitted

We give a lower bound for the polynomial numerical index of order k for real lush spaces. We use this bound to compute the polynomial numerical index of order 2 of the real spaces c 0 , ℓ 1 and ℓ ∞ . Finally, we present an example of a real Banach space X whose polynomial numerical indices are positive while the ones of its bidual are zero.

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.