Some generic fractal properties of bounded self-adjoint operators

TL;DR

This paper investigates the fractal properties of spectral measures of bounded self-adjoint operators, revealing generic behaviors and providing new insights into their dimensional characteristics.

Contribution

It establishes conditions under which spectral measures exhibit generic fractal dimensions and offers a new proof of the Wonderland Theorem for operator spaces.

Findings

Spectral measures typically have extremal fractal dimensions in a generic sense.

Conditions are identified for the prevalence of spectral measures with zero or one fractal dimension.

A new proof of the Wonderland Theorem is provided within the context of operator spaces.

Abstract

We study generic fractal properties of bounded self-adjoint operators through lower and upper generalized fractal dimensions of their spectral measures. Two groups of results are presented. Firstly, it is shown that the set of vectors whose associated spectral measures have lower (upper) generalized fractal dimension equal to zero (one) for every $q>1$ ($0<q<1$) is either empty or generic. The second one gives sufficient conditions, for separable regular spaces of operators, for the presence of generic extreme dimensional values; in this context, we have a new proof of the celebrated Wonderland Theorem.