THERE ARE NO NUMERICAL RADIUS PEAK n-LINEAR MAPPINGS ON c₀

For n ≥ 2 and a real Banach space E, L(ⁿ E: E) denotes the space of all continuous n-linear mappings from E to itself. Let Π(E) = {[x^∗, (x₁, …, x_n)]: x^∗ (x_j) = ∥x^∗ ∥ = ∥x_j ∥ = 1 for j = 1, …, n }. An element [x^∗, (x₁, …, x_n)] ∈ Π(E) is called a numerical radius point of T ∈ L(ⁿ E: E) if |x^∗ (T (x₁, …, x_n))| = v(T), where the numerical radius ∣ v(T) = sup_[y^∗,y₁,…,y_n]∈Π(E) ∣y (T^∗_∣∣. (y₁, …, y_n))∣ For T ∈ L(ⁿ E: E), we define Nradius(T) = {[x^∗, (x₁, …, x_n)] ∈ Π(E): [x^∗, (x₁, …, x_n)] is a numerical radius point of T }. T is called a numerical radius peak n-linear mapping if there is a unique [x^∗, (x₁, …, x_n)] ∈ Π(E) such that Nradius(T) = {±[x^∗, (x₁, …, x_n)]}. In this paper we present explicit formulae for the numerical radius of T for every T ∈ L(ⁿ E: E) for E = c₀ or l_∞. Using these formulae we show that there are no numerical radius peak mappings of L(ⁿ c₀: c₀).