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A formula of local Maslov index and applications

Li Wu, Chaofeng Zhu

TL;DR

This work provides explicit localization for the local Maslov index without symplectic reduction and develops a comprehensive triangular-form framework to compute Maslov-type indices for paths of Lagrangian pairs in symplectic Banach spaces. It combines a structure theorem for triangular decompositions with continuity results for subspaces and linear relations, yielding concrete formulas for the Maslov index, splitting numbers, and iteration-related indices, including mod-2 variants. The methods enable frame-independent analysis of symplectic path iterations and offer practical tools for analyzing infinite-dimensional Maslov-type invariants. Overall, the paper advances both the theory and computational techniques for Maslov-type indices in infinite-dimensional settings and their applications to symplectic dynamics.

Abstract

In this paper, we explicitly express the local Maslov index by a Maslov index in finite dimensional case without symplectic reduction. Then we calculate the Maslov index for the path of pairs of Lagrangian subspaces in triangular form. In particular, we get the Maslov-type index of a given symplectic path in triangle form. As applications, we calculate the splitting numbers of the symplectic matrix in triangle form, dependence of iteration theory on triangular frames and mod 2 Maslov-type index for a real symplectic path. We study the continuity of families of bounded linear relations and families of bounded linear operators acting on closed linear subspaces as technique preparations.

A formula of local Maslov index and applications

TL;DR

This work provides explicit localization for the local Maslov index without symplectic reduction and develops a comprehensive triangular-form framework to compute Maslov-type indices for paths of Lagrangian pairs in symplectic Banach spaces. It combines a structure theorem for triangular decompositions with continuity results for subspaces and linear relations, yielding concrete formulas for the Maslov index, splitting numbers, and iteration-related indices, including mod-2 variants. The methods enable frame-independent analysis of symplectic path iterations and offer practical tools for analyzing infinite-dimensional Maslov-type invariants. Overall, the paper advances both the theory and computational techniques for Maslov-type indices in infinite-dimensional settings and their applications to symplectic dynamics.

Abstract

In this paper, we explicitly express the local Maslov index by a Maslov index in finite dimensional case without symplectic reduction. Then we calculate the Maslov index for the path of pairs of Lagrangian subspaces in triangular form. In particular, we get the Maslov-type index of a given symplectic path in triangle form. As applications, we calculate the splitting numbers of the symplectic matrix in triangle form, dependence of iteration theory on triangular frames and mod 2 Maslov-type index for a real symplectic path. We study the continuity of families of bounded linear relations and families of bounded linear operators acting on closed linear subspaces as technique preparations.
Paper Structure (12 sections, 52 theorems, 289 equations)

This paper contains 12 sections, 52 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. The $\omega$-closed linear subspaces
  3. A formula of local Maslov index without symplectic reduction
  4. Fredholm pairs of Lagrangian subspaces
  5. The Maslov index for the path of pairs of Lagrangian subspaces in triangular form
  6. Applications
  7. Maslov type index
  8. A formula of Maslov-type index
  9. Splitting numbers
  10. Dependence of iteration theory on triangular frames
  11. Mod 2 Maslov-type index
  12. Continuous families of bounded linear relations

Key Result

Theorem 1.1

Let $\varepsilon>0$ be a positive number. Let $X$ be a (complex) Banach space with continuously varying symplectic structure $\omega(s)$, $s\in(-\varepsilon,\varepsilon)$. Let $(\lambda(s),\mu(s))$, $s\in(-\varepsilon,\varepsilon)$ be a path of Fredholm pairs of Lagrangian subspaces of $(X,\omega(s) Then there exists a $\delta\in (0,\varepsilon)$ such that the following hold for all $s\in(-\delta,

Theorems & Definitions (111)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 101 more