Minimal Projections with Respect to Numerical Radius

Abstract

In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases L^p, p = 1, 2, ∞, there is no difference between the minimality of projections measured either with respect to operator norm or with respect to numerical radius. However, we give an example of a projection from lp 3 onto a two-dimensional subspace which is minimal with respect to norm, but not with respect to numerical radius for p ≠ 1, 2,∞. Furthermore, utilizing a theorem of Rudin and motivated by Fourier projections, we give a criterion for minimal projections, measured in numerical radius. Additionally, some results concerning strong unicity of minimal projections with respect to numerical radius are given.