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 $*$-Property. We also characterize the coproximinal subspaces and the co-Chebyshev subspaces of $\mathcal{D}_n$ in terms of the $*$-Property. We observe that a complete characterization of the best coapproximation problem in $\ell_{\infty}^n$ follows directly as a particular case of our approach.