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 of a Banach space. If is a Banach space, let denote the space of the -homogeneous polynomials from to , endowed with its usual sup norm. The numerical radius of a polynomial is given by
The polynomial numerical index of order of is given by
The authors prove some general properties of the polynomial numerical index of a Banach space. For instance, they prove that for every integer and for every complex Banach space . Moreover, they show inequalities between and . If is a scattered compact topological space, then . For the cases or (the disc algebra), the authors prove that for every , where is the symmetric -linear map associated to and