The numerical index of 2-dimensional Lipschitz-free spaces

Abstract

We provide the explicit formula for the numerical index of any 2-dimensional Lipschitz-free space, also giving the construction of operators attaining this value as its numerical radius. As a consequence, the numerical index of 2-dimensional Lipschitz-free spaces can take any value of the interval [12,1], and this whole range of numerical indices can be attained by taking 2-dimensional subspaces of any Lipschitz-free space of the form F(A), where A⊂Rn with n≥2 is any set with non-empty interior.