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Abstract. In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases L p , p = 1, ∞, and in the case of self-adjoint extensions in L 2 , the two extensions and their norms are equal.We also show that, in the case of L p , 1 < p < ∞, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is 1. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from l p 3 onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for p = 1, 2, ∞, and we determine the minimal numerical-radius projection in this same situation.

Let X be a Banach space and a positive measure. In this article, we show that nðL p ð, XÞÞ ¼ lim m nðl m p ðXÞÞ, 1 p < 1. Also, we investigate the positivity of the numerical index of l p -spaces.

Abstract. We present an elementary proof of a general version of Montel's theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu's type theorem for functions f : R d → R for d > 1. Furthermore, our proof of this result is also valid for the case d = 1, differing in several points from Popoviciu's original proof. Finally, we demonstrate that our results are optimal.

Let X be a reflexive Banach space. In this article, we give a necessary and sufficient condition for an operator T ∈ (X ) to have the best approximation in numerical radius from the convex subset ⊂ (X ), where (X ) denotes the set of all linear, compact operators from X into X We also present an application to minimal extensions with respect to the numerical radius. In particular, some results on best approximation in norm are generalized to the case of the numerical radius.