Minimal numerical-radius extensions of operators

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 $L^p, p = 1, \infty$, 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 $L^p$, $1<p<\infty$, 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 $l_3^p$ onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for $p \ne 1, 2, \infty$, and we determine the minimal numerical-radius projection in this same situation.