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Spectral equivalences through nonstandard samplings

Fabrice Nonez

Abstract

The goal of this paper is to introduce a process that generates, given Hilbert space $H$ and symmetric operator $A$, an embedding of $H$ into an $L_2$-space through which $A$ is extended by a multiplication operator. This process will depend on two parameters, the nonstandard sampling and the standard-biased scale. We will use that process to prove diverse versions of the general spectral theorem, showing its appeal. Furthermore, through landmark examples, we will observe that by carefully tweaking the two parameters, we can make the resulting spaces, embeddings and operators quite explicit and natural.

Spectral equivalences through nonstandard samplings

Abstract

The goal of this paper is to introduce a process that generates, given Hilbert space and symmetric operator , an embedding of into an -space through which is extended by a multiplication operator. This process will depend on two parameters, the nonstandard sampling and the standard-biased scale. We will use that process to prove diverse versions of the general spectral theorem, showing its appeal. Furthermore, through landmark examples, we will observe that by carefully tweaking the two parameters, we can make the resulting spaces, embeddings and operators quite explicit and natural.

Paper Structure

This paper contains 17 sections, 32 theorems, 121 equations.

Table of Contents

  1. Overview
  2. Notation and conventions
  3. Samplings and scales
  4. The induced Loeb probability space
  5. The induced pseudometric and the hull space
  6. Self-adjoint operators
  7. Example: Shift operator
  8. Constructing a sampling and a scale
  9. Nature of the hull space
  10. Relation to the Fourier series
  11. Example: Differential operator
  12. Constructing a sampling
  13. Constructing a scale
  14. Characteristics of the Loeb space
  15. Nature of the hull space
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $H$ be a separable infinite-dimensional Hilbert space, and let $A$ be a densely-defined symmetric operator on $H$. Then, there exists a compact metric space $(\Omega,d)$ endowed with a probability measure $\mu$ on its Borel $\sigma$-algebra, a borelian measurable function $m:\Omega\rightarrow\ma

Theorems & Definitions (98)

  • Theorem 1.1: Spectral theorem
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 88 more