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The polynomial numerical index of a Banach space. (English) Zbl 1122.46002
{\it G. Lumer} [Trans. Am. Math. Soc. 100, 29--43 (1961; Zbl 0102.32701)] introduced the concept of the numerical range and numerical radius of a bounded and linear operator defined on a Banach space. The corresponding concepts for bounded continuous mappings defined on the unit ball of a normed space that are also holomorphic on the open unit ball was introduced by {\it L. A. Harris} [Am. J. Math. 93, 1005--1019 (1971; Zbl 0237.58010)]. In the present paper, the authors introduce the polynomial index of order $k$ of a Banach space. If $X$ is a Banach space, let ${\mathcal{P}} (^k X:X)$ denote the space of the $k$-homogeneous polynomials from $X$ to $X$, endowed with its usual sup norm. The numerical radius $v(P)$ of a polynomial $P\in {\mathcal{P}} (^k X:X)$ is given by $$v(P) = \sup \{ \vert x^* (P(x)) \vert : x\in X, x^* \in X^*, \Vert x \Vert = \Vert x ^* \Vert = x^* (x) = 1 \}.$$ The polynomial numerical index of order $k$ of $X$ is given by $$n^{(k)} (X):= \inf \{ v(P): P \in {\mathcal{P}} (^k X:X), \Vert P \Vert = 1 \}.$$ The authors prove some general properties of the polynomial numerical index of a Banach space. For instance, they prove that $ n^{(k)} (X) \ge k^{k(1-k)}$ for every integer $k \ge 2$ and for every complex Banach space $X$. Moreover, they show inequalities between $n^{(k)} (X) $ and $ n^{(k-1)} (X)$. If $K$ is a scattered compact topological space, then $n^{(k)} (C(K))=1$. For the cases $X= C(K)$ or $X= A_D $ (the disc algebra), the authors prove that $ v({\check{P}}) \ge \Vert P \Vert $ for every $P \in {\mathcal{P}} (^k X:X) $, where ${\check{P}}$ is the symmetric $k$-linear map associated to $P$ and $$v( {\check{P}}) = \sup \{ \vert x^* ({\check{P}} (x_1, \dots, x_k))|: x^* \in X^*,\ x_i \in X,\ \Vert x_i \Vert = x^* (x_i) =1, \ \forall i \}.$$

46B04Isometric theory of Banach spaces
46B20Geometry and structure of normed linear spaces
47H60Multilinear and polynomial operators
47A12Numerical range and numerical radius of linear operators
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