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Symmetric preparation of systems

Nils Dencker

TL;DR

This work extends the Weierstrass and Malgrange preparation theorems to symmetric matrix-valued systems. For a smooth or analytic symmetric $N\times N$ matrix function $F(t,x)$ vanishing to first order with $\partial_t F(0,0) > 0$ and $F(0,0)=0$, it establishes the existence of a local symmetric factorization $F(t,x)=U(t,x)\bigl(t\operatorname{Id}_N+M(x)\bigr)U^*(t,x)$ with $|U(0,0)|\neq0$ and $M(x)$ symmetric, $M(0)=0$, and proves uniqueness in the symmetric case ($U$ positive definite). When $F(0,0)\neq0$, the remainder $F(0,0)$ can be accommodated, yielding a version with remainder. The analytic case is proved via an analytic inverse function theorem, while the smooth case uses the Nash–Moser theorem, aided by a symmetric Malgrange division, to obtain locally solvable and tame mappings. Together, these results generalize scalar preparation to symmetric matrix systems, providing a canonical, symmetry-preserving factorization with implications for spectral theory and operator analysis.

Abstract

In this paper we generalize the Weierstrass and Malgrange preparation theorems to the symmetric matrix valued case, proving symmetric preparation of analytic and smooth symmetric systems that vanish of first order.

Symmetric preparation of systems

TL;DR

This work extends the Weierstrass and Malgrange preparation theorems to symmetric matrix-valued systems. For a smooth or analytic symmetric matrix function vanishing to first order with and , it establishes the existence of a local symmetric factorization with and symmetric, , and proves uniqueness in the symmetric case ( positive definite). When , the remainder can be accommodated, yielding a version with remainder. The analytic case is proved via an analytic inverse function theorem, while the smooth case uses the Nash–Moser theorem, aided by a symmetric Malgrange division, to obtain locally solvable and tame mappings. Together, these results generalize scalar preparation to symmetric matrix systems, providing a canonical, symmetry-preserving factorization with implications for spectral theory and operator analysis.

Abstract

In this paper we generalize the Weierstrass and Malgrange preparation theorems to the symmetric matrix valued case, proving symmetric preparation of analytic and smooth symmetric systems that vanish of first order.
Paper Structure (7 sections, 12 theorems, 78 equations)

This paper contains 7 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction and statement of results
  2. Formal power series solutions
  3. Proof of Theorem \ref{['anasymprep']} in the analytic case
  4. Proof of Theorem \ref{['anasymprep']} in the smooth case
  5. A symmetry lemma
  6. Symmetric Malgrange division
  7. Tame Fréchet spaces and the Nash--Moser theorem

Key Result

Theorem 1.2

Assume that $F(t,x)$ is a smooth symmetric $N \times N$ matrix valued function of $(t,x) \in \mathbf R \times \mathbf R^n$ such that Then there exists a neighborhood $\omega$ of $(0,0)$ and smooth $N \times N$ matrix valued functions $U(t,x)$ and $M(x)$ such that $| U(0,0) | \ne 0$, $M(x)$ is symmetric $\forall \, x$, $M(0) = 0$ and If $F(t,x)$ is analytic then we obtain sprepres with analytic $

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Remark 2.2
  • Lemma A.1
  • proof
  • Lemma B.1
  • proof
  • Proposition B.2
  • ...and 17 more