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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( 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 P𝒫( k X:X) is given by

v(P)=sup{|x * (P(x))|:xX,x * X * ,x=x * =x * (x)=1}·

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

n (k) (X):=inf{v(P):P𝒫( k X:X),P=1}·

The authors prove some general properties of the polynomial numerical index of a Banach space. For instance, they prove that n (k) (X)k k(1-k) for every integer k2 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(P ˇ)P for every P𝒫( k X:X), where P ˇ is the symmetric k-linear map associated to P and

v(P ˇ)=sup{|x * (P ˇ(x 1 ,,x k ))|:x * X * ,x i X,x i =x * (x i )=1,i}·


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