Distances between operators acting on different Hilbert spaces
Olaf Post, Sebastian Zimmer
TL;DR
The paper addresses how to compare operators that act on different Hilbert spaces by introducing and relating three distances: the unitary distance $d_{ ext{uni}}$, the isometric distance $d_{ ext{iso}}$, and the quasi-unitary distance $d_{ ext{que}}$, alongside spectral refinements via crude multiplicity functions. It shows that for self-adjoint operators with $0$ in the essential spectrum, $d_{ ext{uni}}$ and $d_{ ext{iso}}$ coincide and that $d_{ ext{que}}$ is equivalent to $d_{ ext{iso}}$ up to a universal constant, with both controlling spectral convergence and Hausdorff spectrum convergence. The work builds on and unifies prior ideas (crude multiplicity, Lévy-Prokhorov distance, Weidmann convergence) and provides sharp examples illustrating the necessity of assumptions; it also connects convergence notions to broader operator theory, including unbounded operators. These results offer a robust framework for analyzing operator convergence across varying spaces, with potential applications to domain perturbations and graph-like structures. Overall, the article advances a metric, spectrum-aware theory for comparing operators across disparate Hilbert spaces, clarifying when different distance notions agree and how they govern spectral behavior. The findings have significant implications for understanding convergent operator families in applied settings where the underlying spaces change with parameters.
Abstract
The aim of this article is to define and compare several distances (or metrics) between operators acting on different (separable) Hilbert spaces. We consider here three main cases of how to measure the distance between two bounded operators: first by taking the distance between their unitary orbits, second by isometric embeddings (this generalises a concept of Weidmann) and third by quasi-unitary equivalence (using a concept of the first author of the present article). Our main result is that the unitary and isometric distances are equal provided the operators are both self-adjoint and have $0$ in their essential spectra. Moreover, the quasi-unitary distance is equivalent (up to a universal constant) with the isometric distance for any pair of bounded operators. The unitary distance gives an upper bound on the Hausdorff distance of their spectrum. If both operators have purely essential spectrum, then the unitary distance equals the Hausdorff distance of their spectra. Using a finer spectral distance respecting multiplicity of discrete eigenvalues, this spectral distance equals the unitary distance also for operators with essential and discrete spectrum. In particular, all operator distances mentioned above are equal to this spectral distance resp. controlled by it in the quasi-unitary case for self-adjoint operators with $0$ in the essential spectrum. We also show that our results are sharp by presenting various (counter-)examples. Finally, we discuss related convergence concepts complementing results from our first article arXiv:2202.03234