Explicit upper bounds for the spectral distance of two trace class operators

TL;DR

This paper derives explicit upper bounds for the spectral distance between two trace class operators using their operator norm difference and singular values, improving existing bounds under certain asymptotics.

Contribution

It introduces new bounds for spectral distances of trace class operators based on determinants and singular value asymptotics, extending previous results.

Findings

Bounds reproduce or improve existing spectral distance bounds

Bounds depend only on operator norm difference and singular values

Proof involves new bounds for determinants of trace class operators

Abstract

Given two trace class operators A and B on a separable Hilbert space we provide an upper bound for the Hausdorff distance of their spectra involving only the distance of A and B in operator norm and the singular values of A and B. By specifying particular asymptotics of the singular values our bound reproduces or improves existing bounds for the spectral distance. The proof is based on lower and upper bounds for determinants of trace class operators of independent interest.