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Gaps and relative dimensions

Chenfeng Liao, Chaofeng Zhu

TL;DR

The paper develops a rigorous framework for relative dimension between closed subspaces under semi-compact perturbations in Banach spaces, introducing right/global semi-compact perturbations and proving that the relative dimension $[M-N]$ is well-defined and stable under appropriate perturbations. It builds a Fredholm-tuple theory (including semi-Fredholm and Fredholm indices) and establishes index formulas for compact perturbations, along with stability results in both complemented and noncomplemented settings. An embedding technique into spaces with the approximation property is used to handle general domains, enabling global-stability results for semi-compact perturbations. The work culminates with stability theorems for semi-Fredholm tuples and a study of the perturbed augmented Morse index, providing concrete bounds and duality results that extend classical Morse theory to this functional-analytic setting. Overall, the results offer a robust toolkit for measuring and preserving relative-dimensional information under perturbations in Banach spaces, with potential applications to spectral theory and variational analysis.

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.

Gaps and relative dimensions

TL;DR

The paper develops a rigorous framework for relative dimension between closed subspaces under semi-compact perturbations in Banach spaces, introducing right/global semi-compact perturbations and proving that the relative dimension is well-defined and stable under appropriate perturbations. It builds a Fredholm-tuple theory (including semi-Fredholm and Fredholm indices) and establishes index formulas for compact perturbations, along with stability results in both complemented and noncomplemented settings. An embedding technique into spaces with the approximation property is used to handle general domains, enabling global-stability results for semi-compact perturbations. The work culminates with stability theorems for semi-Fredholm tuples and a study of the perturbed augmented Morse index, providing concrete bounds and duality results that extend classical Morse theory to this functional-analytic setting. Overall, the results offer a robust toolkit for measuring and preserving relative-dimensional information under perturbations in Banach spaces, with potential applications to spectral theory and variational analysis.

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.
Paper Structure (11 sections, 41 theorems, 210 equations)

This paper contains 11 sections, 41 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Stability theorems for semi-compact perturbations
  3. Fredhom pairs of closed linear subspaces
  4. Riesz-Schauder theory
  5. Relative dimension between closed linear subspaces
  6. An index formula for compact perturbations
  7. Stability theorem for semi-compact perturbation of complemented linear subspaces
  8. An isometric linear embedding of a Banach space into some Banach space with (AP)
  9. Stability theorem for global semi-compact perturbation
  10. Stability theorems for semi-Fredholm tuples
  11. Perturbed augmented Morse index

Key Result

Theorem 1.1

Let $X$ be a Banach space with two closed linear subspaces $M$ and $N$. Assume that $N$ is a right global semi-compact perturbation of $M$. Then the following hold.

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Definition 2.1: The gap between linear subspaces
  • Proposition 2.2: Finite extension
  • ...and 80 more