Geometry of the space of compact operators endowed with the numerical radius norm

We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on ${\mathrm{\ell }}_p$ $(1<p<\mathrm{\infty })$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of ${\mathrm{\ell }}_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of ${\mathrm{\ell }}_1$ whose numerical index is greater than $1\mathrm{/}2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius.