Conjecture 1.7: Existence of arbitrarily close globe compact perturbations with zero relative dimension

Establish the existence of a closed linear subspace M of an arbitrary Banach space X such that, for every epsilon > 0, there exists a closed linear subspace N of X with the properties that N is a globe compact perturbation of M (i.e., there exists a compact operator K in B(X) such that (I + K)M is contained in N and the map I + K: M → N is Fredholm), the gap distance δ(M, N) is less than epsilon, and the relative dimension [M − N] equals 0.

Background

The authors introduce stability properties for indices and relative dimensions of pairs of closed subspaces under various perturbations. For semi-compact perturbations, their main stability theorem (Theorem 1.6) requires an approximation property (AP) on the compact operator involved, due to the lack of direct control over operator norms when comparing perturbations. To address this limitation, they propose a conjecture guaranteeing the existence of subspaces that admit arbitrarily close globe compact perturbations with zero relative dimension, which would conceptually bypass the need for (AP) in certain stability arguments.

This conjecture concerns the gap topology on the Grassmannian of closed subspaces of a Banach space and the index-theoretic relative dimension defined via compact perturbations. It posits that for some closed subspace M, one can approximate M arbitrarily well by closed subspaces N connected to M via globe compact perturbations with trivial relative dimension, i.e., [M − N] = 0.