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Skew-adjoint linear relatioins between Banach spaces

Hanchen Li, Chaofeng Zhu

TL;DR

This work extends the theory of skew-adjoint and selfadjoint linear relations from Hilbert spaces to Banach spaces by developing a robust symplectic Banach framework. It establishes stability theorems for maximal isotropic subspaces under perturbations of the symplectic form, including Morse-index perturbations and relative bounded perturbations, with simplifications in the strong symplectic setting. It then analyzes real Banach-space skew-adjoint Fredholm relations of index $0$, proving a mod $2$ stability result for the kernel and describing the exact two path components of the relation space, using a real–complex correspondence and Witt-type decompositions. Collectively, these results generalize known Hilbert-space phenomena (e.g., CCGP23, Hess–Kato) to the Banach setting and provide a foundation for oriented flow constructions and topological classifications of skew-adjoint relations in Banach spaces.

Abstract

In this paper, we prove the stability theorems for the isotropic perturbations of maximal isotropic subspaces in symplectic Banach spaces. Then we prove a stability theorem for the mod $2$ dimensions of kernel of skew-adjoint linear Fredholm relations between real Banach spaces with index $0$. Finally we gives the two path components of the set of skew-adjoint linear Fredholm relations between real Banach spaces with indices $0$.

Skew-adjoint linear relatioins between Banach spaces

TL;DR

This work extends the theory of skew-adjoint and selfadjoint linear relations from Hilbert spaces to Banach spaces by developing a robust symplectic Banach framework. It establishes stability theorems for maximal isotropic subspaces under perturbations of the symplectic form, including Morse-index perturbations and relative bounded perturbations, with simplifications in the strong symplectic setting. It then analyzes real Banach-space skew-adjoint Fredholm relations of index , proving a mod stability result for the kernel and describing the exact two path components of the relation space, using a real–complex correspondence and Witt-type decompositions. Collectively, these results generalize known Hilbert-space phenomena (e.g., CCGP23, Hess–Kato) to the Banach setting and provide a foundation for oriented flow constructions and topological classifications of skew-adjoint relations in Banach spaces.

Abstract

In this paper, we prove the stability theorems for the isotropic perturbations of maximal isotropic subspaces in symplectic Banach spaces. Then we prove a stability theorem for the mod dimensions of kernel of skew-adjoint linear Fredholm relations between real Banach spaces with index . Finally we gives the two path components of the set of skew-adjoint linear Fredholm relations between real Banach spaces with indices .
Paper Structure (12 sections, 32 theorems, 159 equations)

This paper contains 12 sections, 32 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Stability theorems in symplectic Banach space
  3. Fredholm pairs of closed linear subspaces
  4. Symplectic Banach spaces
  5. Perturbed Morse index
  6. Stability theorems of maximal isotropic subspaces
  7. Stability theorems in strong symplectic Banach spaces
  8. Stabilities with relative bounded perturbations
  9. Skew-adjoint relations between real Banach spaces
  10. Skew-adjoint relations between real Hilbert spaces
  11. Stability theorem of mod 2 index
  12. Path components

Key Result

Theorem 1.1

Let $(X,\omega_0)$ be a symplectic Banach space with a maximal isotropic subspace $\lambda$. We assume that $\gamma_\lambda>0$ holds if $h_\lambda\ne 0$. Let $(X,\omega)$ be a symplectic Banach space with a closed isotropic subspace of $\mu$. We set Here $h_\lambda$ and $\gamma_\lambda$ are defined by Definition d:sign-max-isotroic below, and $l(\mu,\omega)=0$ if $h_\lambda=0$. Assume that there

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1: The gap between closed linear subspaces
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 66 more