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Representing Locally Hilbert Spaces and Functional Models for Locally Normal Operators

Aurelian Gheondea

TL;DR

This work develops a spectral theory for locally normal operators on representing locally Hilbert spaces by embedding their analysis within strictly inductive systems of measure spaces. It constructs functional models as projective limits of operators multiplying by locally $L^\infty$-type functions and obtains a triad of spectral theorems: a First spectral theorem via locally spectral measures, a Second spectral theorem with a multiplication-by-local-L^\infty model under a sequentially finite directed set, and a Third spectral theorem via direct integral representations. The Hata tree-like self-similar set serves as a concrete fractal example to illustrate the inductive-measure-space framework and the multiplicity data encoded in pseudo-concrete models. These results connect spectral theory on locally Hilbert spaces with Abelian von Neumann algebras and open avenues for fractal-analysis-inspired operator theory on nonstandard spaces.

Abstract

The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem in terms of projective limits of certain multiplication operators with functions which are locally of type $L^\infty$. In order to do this, we first investigate strictly inductive systems of measure spaces and point out the concept of representing locally Hilbert space for which we obtain a functional model as a strictly inductive limit of $L^2$ type spaces. Then, we first obtain a functional model for locally normal operators on representing locally Hilbert spaces combined with a spectral multiplicity model on a pseudo-concrete functional model for the underlying locally Hilbert space, under a certain technical condition on the directed set. Finally, under the same technical condition on the directed set, we derive the spectral theorem for locally normal operators in terms of projective limits of certain multiplication operators with functions which are locally of type $L^\infty$ in two forms. As a consequence of the main result we sketch the direct integral representation of locally normal operators under the same technical assumptions and the separability of the locally Hilbert space. Examples of strictly inductive systems of measure spaces involving the Hata tree-like self-similar set, which justify the technical condition on a relevant case and which may open a connection with analysis on fractal sets, are included.

Representing Locally Hilbert Spaces and Functional Models for Locally Normal Operators

TL;DR

This work develops a spectral theory for locally normal operators on representing locally Hilbert spaces by embedding their analysis within strictly inductive systems of measure spaces. It constructs functional models as projective limits of operators multiplying by locally -type functions and obtains a triad of spectral theorems: a First spectral theorem via locally spectral measures, a Second spectral theorem with a multiplication-by-local-L^\infty model under a sequentially finite directed set, and a Third spectral theorem via direct integral representations. The Hata tree-like self-similar set serves as a concrete fractal example to illustrate the inductive-measure-space framework and the multiplicity data encoded in pseudo-concrete models. These results connect spectral theory on locally Hilbert spaces with Abelian von Neumann algebras and open avenues for fractal-analysis-inspired operator theory on nonstandard spaces.

Abstract

The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem in terms of projective limits of certain multiplication operators with functions which are locally of type . In order to do this, we first investigate strictly inductive systems of measure spaces and point out the concept of representing locally Hilbert space for which we obtain a functional model as a strictly inductive limit of type spaces. Then, we first obtain a functional model for locally normal operators on representing locally Hilbert spaces combined with a spectral multiplicity model on a pseudo-concrete functional model for the underlying locally Hilbert space, under a certain technical condition on the directed set. Finally, under the same technical condition on the directed set, we derive the spectral theorem for locally normal operators in terms of projective limits of certain multiplication operators with functions which are locally of type in two forms. As a consequence of the main result we sketch the direct integral representation of locally normal operators under the same technical assumptions and the separability of the locally Hilbert space. Examples of strictly inductive systems of measure spaces involving the Hata tree-like self-similar set, which justify the technical condition on a relevant case and which may open a connection with analysis on fractal sets, are included.

Paper Structure

This paper contains 24 sections, 27 theorems, 205 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Inductive and Projective Limits
  2. Projective Limits.
  3. Inductive Limits.
  4. Coherent Maps between Inductive Limits.
  5. Coherent Maps between Projective Limits
  6. Locally Hilbert Spaces and Their Linear Operators
  7. General Properties
  8. Locally Bounded Operators.
  9. Locally $C^*$-Algebras.
  10. The Locally $C^*$-Algebra ${\mathcal{B}}_{\mathrm{loc}}({\mathcal{H}})$.
  11. Representing Locally Hilbert Spaces
  12. Why Representing Locally Hilbert Spaces?
  13. Strictly Inductive Systems of Measurable Spaces.
  14. Strictly Inductive Systems of Measure Spaces.
  15. Concrete Representing Locally Hilbert Spaces
  16. ...and 9 more sections

Key Result

Proposition 2.3

Let ${\mathcal{S}}$ be a linear subspace of ${\mathcal{H}}$. The following assertions are equivalent.

Figures (2)

  • Figure 1: The $14$th approximation in generating the Hata tree-like self-similar set, for $c=0.3+0.4\mathrm{i}$ and the seed $X_0=[0,1]\cup c[0,1]$, provides an idea of the tree form of this self-similar set. The picture is obtained with a MATLAB code based on Algorithm 1.
  • Figure 2: The seed $X_0=[0,1]\cup c[0,1]$ and the first five approximation steps in generating the Hata tree-like self-similar set, for $c=0.3+0.4\mathrm{i}$. $X_0$ has $2=2^0+1$ branches, $X_1$ has $3=2^1+1$ branches, $X_2$ has $5=2^2+1$ branches, $X_3$ has $9=2^3+1$ branches, $X_4$ has $17=2^4+1$ branches, and $X_5$ has $33=2^5+1$ branches. The pictures are obtained with a MATLAB code based on Algorithm 1.

Theorems & Definitions (55)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 45 more