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:is sharp.(iii) The inequalities(iv) The relation between the polynomial numerical index of c 0 , l 1 , l∞ 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 * * ) 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.

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It is well-known that the image of a multilinear mapping into a vector space need not be a subspace of its target space. It is, however, far from clear which subsets of the target space may be such images. For vector spaces over the real numbers we give a complete classification of the images of bilinear mappings into a three-dimensional vector space. In Theorem 2.8 we show that either the image of a bilinear mapping into a three-dimensional space is a subspace, or its complement is either the interior of a double elliptic cone, or a plane from which two lines intersecting at the origin have been removed. We also show (Theorem 2.2) that the image of any multilinear mapping into a two-dimensional space is necessarily a subspace. Our methods are elementary and free of tensor considerations.

Globevnik gave the definition of boundary for a subspace A ⊂ C b (Ω). This is a subset of Ω that is a norming set for A. We introduce the concept of numerical boundary. For a Banach space X, a subset B ⊂ Π(X) is a numerical boundary for a subspace A ⊂ C b (B X , X) if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in A. We give examples of numerical boundaries for the complex spaces X = c 0 , C(K) and d * (w, 1), the predual of the Lorentz sequence space d(w, 1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from B X to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c 0 , we characterize the numerical boundaries for that space of holomorphic functions.

For two complex Banach spaces X and Y , A∞(B X ; Y ) will denote the space of bounded and continuous functions from B X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in A∞(B X ; X) is the supremum of the setWe prove that every complex Banach space X with the Radon-Nikodým property satisfies that the subset of numerical radius attaining functions in A∞(B X ; X) is dense in A∞(B X ; X). We also show the denseness of the numerical radius attaining elements of Au(Bc 0 ; c 0 ) in the whole space, where Au(Bc 0 ; c 0 ) is the subset of functions in A∞(Bc 0 ; c 0 ) which are uniformly continuous on the unit ball. For C(K) we prove a denseness result for the subset of the functions in A∞(B C(K) ; C(K)) which are weakly uniformly continuous on the closed unit ball. For a certain sequence space X, there is a 2-homogenous polynomial P from X to X such that for every R > e, P cannot be approximated by bounded and 373 374 M. D. ACOSTA AND S. G. KIM Isr. J. Math.numerical radius attaining holomorphic functions defined on RB X . If Y satisfies some isometric conditions and X is such that the subset of norm attaining functions of A∞(B X ; C) is dense in A∞(B X ; C), then the subset of norm attaining functions in A∞(B X ; Y ) is dense in the whole space.