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The polynomial numerical index of a Banach space. (English) Zbl 1122.46002

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 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 $𝒫{\left(}^{k}X:X\right)$ denote the space of the $k$-homogeneous polynomials from $X$ to $X$, endowed with its usual sup norm. The numerical radius $v\left(P\right)$ of a polynomial $P\in 𝒫{\left(}^{k}X:X\right)$ is given by

$v\left(P\right)=sup\left\{|{x}^{*}\left(P\left(x\right)\right)|:x\in X,{x}^{*}\in {X}^{*},\parallel x\parallel =\parallel {x}^{*}\parallel ={x}^{*}\left(x\right)=1\right\}·$

The polynomial numerical index of order $k$ of $X$ is given by

${n}^{\left(k\right)}\left(X\right):=inf\left\{v\left(P\right):P\in 𝒫{\left(}^{k}X:X\right),\parallel P\parallel =1\right\}·$

The authors prove some general properties of the polynomial numerical index of a Banach space. For instance, they prove that ${n}^{\left(k\right)}\left(X\right)\ge {k}^{k\left(1-k\right)}$ for every integer $k\ge 2$ and for every complex Banach space $X$. Moreover, they show inequalities between ${n}^{\left(k\right)}\left(X\right)$ and ${n}^{\left(k-1\right)}\left(X\right)$. If $K$ is a scattered compact topological space, then ${n}^{\left(k\right)}\left(C\left(K\right)\right)=1$. For the cases $X=C\left(K\right)$ or $X={A}_{D}$ (the disc algebra), the authors prove that $v\left(\stackrel{ˇ}{P}\right)\ge \parallel P\parallel$ for every $P\in 𝒫{\left(}^{k}X:X\right)$, where $\stackrel{ˇ}{P}$ is the symmetric $k$-linear map associated to $P$ and

$v\left(\stackrel{ˇ}{P}\right)=sup\left\{|{x}^{*}\left(\stackrel{ˇ}{P}\left({x}_{1},\cdots ,{x}_{k}\right)\right)|:{x}^{*}\in {X}^{*},\phantom{\rule{4pt}{0ex}}{x}_{i}\in X,\phantom{\rule{4pt}{0ex}}\parallel {x}_{i}\parallel ={x}^{*}\left({x}_{i}\right)=1,\phantom{\rule{4pt}{0ex}}\forall i\right\}·$

##### MSC:
 46B04 Isometric theory of Banach spaces 46B20 Geometry and structure of normed linear spaces 47H60 Multilinear and polynomial operators 47A12 Numerical range and numerical radius of linear operators