On best coapproximations in subspaces of diagonal matrices

TL;DR

This paper characterizes the best coapproximations in subspaces of diagonal matrices using Birkhoff-James orthogonality and introduces the *-Property, providing a comprehensive framework for coapproximation problems.

Contribution

It introduces the *-Property and applies it to characterize best coapproximations, coproximinal, and co-Chebyshev subspaces in diagonal matrices, extending to \,\ell_{\infty}^n.

Findings

Characterization of best coapproximations using the *-Property.

Complete description of coproximinal and co-Chebyshev subspaces.

Application of the approach to \ell_{\infty}^n cases.

Abstract

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.