Transfer of Approximation properties under Local Constraints and Best Simultaneous Approximation on Sums

TL;DR

This paper investigates how approximation properties in Banach spaces, especially involving sums of subspaces, are preserved or fail, extending previous results and addressing open problems in the field.

Contribution

It explores the preservation of approximation properties under sums of subspaces, provides counterexamples, and extends results to best simultaneous approximation properties.

Findings

Counterexamples show non-preservation of properties $(GC)$ and the central subspace.

The paper answers an open problem from a 2015 publication.

Extends observations to properties $(P_1)$ and $ ext{F}$-SACP.

Abstract

It is folklore that the sum of two $M$-ideals (semi $M$-ideals) is also an $M$-ideal (a semi $M$-ideal). Numerous authors have attempted to investigate such properties of subspaces. This article explores two important facets of approximation theory within Banach spaces and how these properties remain intact when considering the sum of two subsets. Recall the notion of $(GC)$ introduced by Vesel\'y that encloses two aforementioned properties. When the sum of two subspaces is closed, we discuss various properties of the sum if one of the subspaces has these properties. Counterexamples are produced that establish nonaffirmativeness for the properties $(GC)$ and the central subspace. We answer a problem raised by the author in [{\em Best constrained approximation in Banach spaces}, Numer. Funct. Anal. Optim. {\bf 36}(2) (2015), 248--255]. We extend our observations related to the best simultaneous approximations to the properties $(P_1)$ and $\mr{F}$-SACP.