Quasi-unitary equivalence and generalised norm resolvent convergence

TL;DR

This work develops a generalised norm-resolvent framework based on $δ$-quasi-unitary equivalence to compare non-negative self-adjoint operators across different Hilbert spaces, yielding explicit transitivity and convergence-rate results. It shows how operator and energy-form versions interrelate and how holomorphic functional calculus, spectral projections, and heat semigroups are stable under these perturbations, providing quantitative bounds. The framework is illustrated through fractal and obstacle examples: unit interval and Sierpiński gasket approximated by finite graphs, with explicit convergence rates, and Neumann-obstacle perturbations on manifolds, highlighting spectral convergence and eigenfunction transfer. These results have potential implications for homogenisation, space-perturbation problems, and limits of manifold-graph and graph-like structures where the underlying space changes.

Abstract

The purpose of this article is to give a short introduction to the concept of quasi-unitary equivalence of quadratic forms and its consequences. In particular, we improve an estimate concerning the transitivity of quasi-unitary equivalence for forms. We illustrate the abstract setting by two classes of examples.

Theorems & Definitions (17)

• Definition 1.1
• Definition 1.2
• Proposition 1.3: post:12
• Definition 1.4
• Proposition 1.5
• Proposition 1.6
• Lemma 1.7: post-simmer:18
• Theorem 1.8
• Proposition 1.9: post:12
• Example 1.10