On best coapproximations in subspaces of diagonal matrices

We characterize the best coapproximation(s) to a given matrix $T$ out of a given subspace $\mathbb{Y}$ of the space of diagonal matrices ${\mathcal{D}}_n$, by using Birkhoff-James orthogonality techniques and with the help of a newly introduced property, christened the $\ast$-Property. We also characterize the coproximinal subspaces and the co-Chebyshev subspaces of ${\mathcal{D}}_n$ in terms of the $\ast$-Property. We observe that a complete characterization of the best coapproximation problem in ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ follows directly as a particular case of our approach.