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A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators

Takashi Matsunaga

Abstract

By nonstandard analysis, a very short and elementary proof of the Spectral Theorem for unbounded self-adjoint operators is given.

A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators

Abstract

By nonstandard analysis, a very short and elementary proof of the Spectral Theorem for unbounded self-adjoint operators is given.
Paper Structure (6 sections, 14 theorems, 23 equations)

This paper contains 6 sections, 14 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Reminders
  3. Preliminaries
  4. A nonstandard proof the Spectral theorem
  5. Concluding Remarks
  6. Appendix: a quick introduction to nonstandard analysis

Key Result

Lemma 2.2

Suppose that $E(\lambda)$ is a spectral family on a complex Hilbert space $H$ and $f(\lambda)$ is continuous on $\mathbb{R}$. Let $D= \{x \in H | \int_{-\infty}^{\infty} f(\lambda)^2 d(E(\lambda)x, x) < \infty \}$ (integrator: $d(E(\lambda)x, x)$). Then for $a<b \in \mathbb{R}$, the Riemann-Stieltje exist. $D$ is dense in $H$. If $f(\lambda)$ is real-valued then $T$ is self-adjoint. Let $E(\lambda

Theorems & Definitions (40)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • ...and 30 more