The Polynomial Numerical Index of a Banach Space

Abstract

In 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:is sharp.(iii) The inequalities(iv) The relation between the polynomial numerical index of c 0 , l 1 , l∞ 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 * * ) 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.

Polynomial Numerical Index for Some Complex Vector-Valued Function Spaces

“…In [7,Proposition 2.8], it is shown that for every k ∈ N the polynomial numerical index of order k of a c 0 -, 1 -, or ∞ -sum of real or complex Banach spaces is less than or equal to the infimum of those indices of the summands. This result is also true for p -sums, and also for absolute sums.…”

“…The proof for the cases p = 1, ∞ appears in [7,Proposition 2.8]. That proof is easy to extend to an arbitrary absolute sum.…”

“…In [7,Proposition 2.10], it was shown that n (k) (C(K, X)) n (k) (X) for real or complex spaces. This, together with the above result gives the equality in the complex case.…”

“…It was extended to the polynomial case in [7] by considering homogeneous polynomials instead of linear operators. That is, for k ∈ N, we define n (k) (X) = inf{v(P ) : P ∈ P( k X; X), P = 1}, where P( k X; X) is the space of all continuous k-homogeneous polynomials from X into X, and call it the polynomial numerical index of order k of X.…”

“…That is, for k ∈ N, we define n (k) (X) = inf{v(P ) : P ∈ P( k X; X), P = 1}, where P( k X; X) is the space of all continuous k-homogeneous polynomials from X into X, and call it the polynomial numerical index of order k of X. For more information and background, we refer the reader to the recent papers [17] (a survey) and to [7,9,19,20] and references therein.…”