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THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

Published online by Cambridge University Press:  02 February 2006

Yun Sung Choi
Affiliation:
Department of Mathematics, POSTECH, Pohang 790-784, South Korea ([email protected])
Domingo Garcia
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain ([email protected])
Sung Guen Kim
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea ([email protected])
Manuel Maestre
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain ([email protected])
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Abstract

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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:

(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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2006