Best Approximation in Numerical Radius

Abstract

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.

Minimal Projections with Respect to Numerical Radius

“…A natural extension of the above-mentioned results to the case of numerical radius • w was given in [2].…”

“…For Kolmogorov type criteria concerning approximation with respect to numerical radius, we refer the reader to [2].…”

Partial smoothness of the numerical radius at matrices whose fields of values are disks

“…Research on the topic remains very active. 1 An elegant proof of the power inequality relies on the following semidefinite representation of the numerical radius due to [29], based in part on [2]. We denote the space of n-by-n Hermitian matrices by H n .…”

Codimension-one minimal extensions onto Haar subspaces

“…Basic information about minimal projections and extensions one can find in [1,[4][5][6]9,10,15,16,18,21,22]. Let W be a Banach space and V its closed linear subspace.…”

“…If |k| < |l| < 2|k|, by (20), 3 4 l − 1 4 |l| α 3 4 l + 1 4 |l| and 3l − 2|l| α 3l + 2|l|. For l > 0 we can rewrite the above inequality in the form 1 2 l α l and l α 5l, for l < 0 in the form l α 1 2 l and 5l α l. Hence in both cases we conclude that α = l. By (21),…”