Gaps and relative dimensions

TL;DR

This paper introduces the concept of semi-compact perturbations of closed linear subspaces in Banach spaces, establishing the stability of relative dimensions and Fredholm tuples under such perturbations, and explores the perturbed augmented Morse index.

Contribution

It defines semi-compact perturbations, proves the stability of relative dimensions and Fredholm tuples, and studies the perturbed augmented Morse index in Banach spaces.

Findings

Relative dimension is well-defined for semi-compact perturbations.

Stability of relative dimension under global semi-compact perturbations.

Stability of Fredholm tuples and analysis of the perturbed augmented Morse index.

Abstract

In this paper, the notion of semi-compact perturbation of a closed linear subspace is introduced. Then for a of pair of closed linear subspace of a Banach space such that one is a semi-compact perturbation of the other, it is proved that the relative dimension between them is well-defined. If the perturbation is global, the relative dimension is stable, even the perturbed pair is a semi-compact perturbed one. After that, the notion of Fredholm tuple of closed linear subspaces in a Banach space is introduced. Then the stability of the Fredholm tuple is proved. Finally the perturbed augmented Morse index is studied.