[Paper Review] Fractal scale Hilbert spaces and scale Hessian operators

Urs Frauenfelder|ArXiv.org|Dec 7, 2009

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper introduces fractal scale Hilbert spaces—scale Hilbert spaces scale isomorphic to a specific class of weighted $β^2$ spaces defined by an unbounded, monotone function $f$—and establishes that scale Hessian operators on such spaces are characterized by their associated function $f_A$, leading to a complete classification of these spaces via the spectral data of the operator. The key result is that any scale Hessian operator induces a fractal scale Hilbert space structure, and all such spaces are scale isomorphic to $\ell^{2,f_A}$, providing a canonical model for Floer theory setups.

ABSTRACT

Scale spaces were defined by H.Hofer, K.Wysocki, and E.Zehnder. In this note we introduce a subclass of scale spaces and explain why we believe that this subclass is the right class for a general setup of Floer theory.

Motivation & Objective

• To define and characterize a subclass of scale Hilbert spaces—fractal scale Hilbert spaces—that are suitable for a general setup of Floer theory.
• To establish that scale Hessian operators on scale Hilbert spaces induce a canonical structure via their associated function $f_A$, derived from the eigenvalues of the operator.
• To prove that any scale Hilbert space equipped with a scale Hessian operator is scale isomorphic to $\ell^{2,f_A}$, thereby classifying such spaces via spectral data.
• To show that the shift of a fractal scale Hilbert space remains scale isomorphic to the original, highlighting self-similarity in the scale structure.

Proposed method

• Define fractal scale Hilbert spaces as those scale isomorphic to $\ell^{2,f}$ for an unbounded, monotone function $f: \mathbb{N} \to (0,\infty)$, using weighted $\ell^2$ spaces with inner products $\langle x,y\rangle_f = \sum f(\nu)x_\nu y_\nu$.
• Introduce scale Hessian operators as symmetric, unbounded, cocompact, and Fredholm operators on scale Hilbert spaces, satisfying regularity and Fredholm axioms.
• Prove that for a cocompact selfadjoint operator $A$ on $H_0$, the pair $(H_0, \mathrm{dom}(A))$ is scale isometric to $(\ell^2, \ell^2_{f_A})$ with $f_A(\nu) = 1 + \gamma_\nu^2$, where $\gamma_\nu$ are the eigenvalues of $A$.
• Use spectral decomposition to construct a common orthonormal basis across all levels $H_k$, ensuring compatibility with the scale structure.
• Show that the restriction of a scale Hessian operator to $\mathcal{H}^2$ remains a scale Hessian operator on $\mathcal{H}^1$, preserving Fredholm and regularity properties.
• Establish that the entire scale Hilbert space $\mathcal{H}$ is scale isomorphic to $\ell^{2,f_A}$ by constructing an explicit isomorphism via rescaling the eigenbasis by $f_A(\nu)^{-k/2}$ at level $k$.

Experimental results

Research questions

• RQ1 What class of scale Hilbert spaces is most suitable for a general setup of Floer theory, and how can it be characterized?
• RQ2 How do scale Hessian operators on scale Hilbert spaces relate to the underlying spectral data of the operator?
• RQ3 Can every scale Hilbert space equipped with a scale Hessian operator be classified via a single function derived from the operator’s spectrum?
• RQ4 Is the scale isomorphism class of a scale Hilbert space determined solely by the spectral data of a scale Hessian operator acting on it?
• RQ5 What structural properties—such as self-similarity under shifting—characterize fractal scale Hilbert spaces?

Key findings

• Fractal scale Hilbert spaces are defined as scale Hilbert spaces that are scale isomorphic to $\ell^{2,f}$ for some unbounded, monotone function $f: \mathbb{N} \to (0,\infty)$, providing a canonical model for scale structures.
• Every scale Hessian operator on a scale Hilbert space induces a function $f_A(\nu) = 1 + \gamma_\nu^2$, where $\gamma_\nu$ are the eigenvalues of the operator, which fully characterizes the scale structure.
• The scale Hilbert pair $(H_0, \mathrm{dom}(A))$ for a cocompact selfadjoint operator $A$ is scale isometric to $(\ell^2, \ell^2_{f_A})$, establishing a spectral classification.
• The restriction of a scale Hessian operator to $\mathcal{H}^2$ remains a scale Hessian operator on $\mathcal{H}^1$, preserving all axioms including the Fredholm property.
• The entire scale Hilbert space $\mathcal{H}$ is scale isomorphic to $\ell^{2,f_A}$, with the isomorphism constructed explicitly via rescaling the eigenbasis by $f_A(\nu)^{-k/2}$ at level $k$, ensuring compatibility across all scales.
• The shift $\mathcal{H}^m$ of a fractal scale Hilbert space $\mathcal{H}$ is scale isomorphic to $\mathcal{H}$ for all $m \in \mathbb{N}_0$, demonstrating self-similarity in the scale structure.