The Formula for the Quasicentral Modulus in the Case of Spectral Measures on Fractals

TL;DR

This paper establishes a formula for the quasicentral modulus of operator n-tuples with spectra on fractals, linking spectral measures, Lorentz norms, and Hausdorff measures in a novel way.

Contribution

It introduces a general ampliation homogeneity result and derives a new formula connecting spectral measures on fractals with the quasicentral modulus.

Findings

Proves a homogeneity result for the quasicentral modulus.

Derives a formula involving Hausdorff measure for spectral measures on fractals.

Links spectral properties of operators to fractal geometry.

Abstract

We prove a general ampliation homogeneity result for the quasicentral modulus of an n-tuple of operators with respect to the (p,1) Lorentz normed ideal. We use this to prove a formula involving Hausdorff measure for the quasicentral modulus of n-tuples of commuting Hermitian operators the spectrum of which is contained in certain Cantor-like self-similar fractals.