Abstract

The Spectral Embedding Theorem, one of the formulations of the Spectral Theorem, states that any densely-defined symmetric operator A on a separable Hilbert space H can be extended by a multiplication operator through an isometric embedding of H in an L_2-space.

In this research project, the goal is to study a novel process. This process starts with a (real or complex) separable Hilbert space H and a densely-defined symmetric operator A. It results with a compact metric space Ω, a probability measure μ on Ω, an isometric embedding U : H ⟶ L_2(Ω, μ) and a multiplication operator T on L_2(Ω, μ) such that U ∘ A ⊂ T ∘ U, satisfying the Spectral Embedding Theorem.

The process uses two parameters, which are objects of Nonstandard Analysis: the nonstandard sampling and the standard-biased scale. We see in this Thesis that these objects allow for a new proof of the Spectral Embedding Theorem, as well as other forms of the Spectral Theorem.

Furthermore, we will observe that with specific operators, the process can result in explicit and natural objects through the careful tweaking of the parameters. Specifically, we work with landmark examples of the theory, the shift operator and the differential operator on the line, to derive the Fourier series and the Fourier Transform from the process.