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A distance between operators acting in different Hilbert spaces and operator convergence

Olaf Post, Jan Simmer

TL;DR

The article introduces quasi-unitary equivalence and several operator-distance notions for self-adjoint operators that may act on different Hilbert spaces, enabling generalized norm resolvent convergence via identification operators. It develops spectral and quasi-unitary distances, proves how δ-quasi-unitary equivalence yields convergence of spectra and eigenfunctions, and applies the framework to Laplacians on thin branched manifolds converging to a Kirchhoff Laplacian on a metric graph, with resolvent error bounds of order $O( ext{ε}^{1/2})$ and eigenvalue convergence of order $O( ext{ε})$ in related results. This work provides a rigorous mechanism to compare operators across varying spaces, bridging graph and manifold Laplacians and offering a foundation for future refinement of identification maps to improve convergence rates. The methods fuse generalized resolvent convergence with spectrum-sensitive metrics, yielding practical implications for approximations of continuous operators by discrete or lower-dimensional models.

Abstract

The aim of the present article is to give an introduction to the concept of quasi-unitary equivalence and to define several (pseudo-)metrics on the space of self-adjoint operators acting possibly in different Hilbert spaces. As some of the ``metrics'' do not fulfill all properties of a metric (e.g. some lack the triangle inequality or the definiteness), we call them ``distances'' here. To the best of our knowledge, such distances are treated for the first time here. The present article shall serve as a starting point of further research (see e.g. arXiv:2412.13165).

A distance between operators acting in different Hilbert spaces and operator convergence

TL;DR

The article introduces quasi-unitary equivalence and several operator-distance notions for self-adjoint operators that may act on different Hilbert spaces, enabling generalized norm resolvent convergence via identification operators. It develops spectral and quasi-unitary distances, proves how δ-quasi-unitary equivalence yields convergence of spectra and eigenfunctions, and applies the framework to Laplacians on thin branched manifolds converging to a Kirchhoff Laplacian on a metric graph, with resolvent error bounds of order and eigenvalue convergence of order in related results. This work provides a rigorous mechanism to compare operators across varying spaces, bridging graph and manifold Laplacians and offering a foundation for future refinement of identification maps to improve convergence rates. The methods fuse generalized resolvent convergence with spectrum-sensitive metrics, yielding practical implications for approximations of continuous operators by discrete or lower-dimensional models.

Abstract

The aim of the present article is to give an introduction to the concept of quasi-unitary equivalence and to define several (pseudo-)metrics on the space of self-adjoint operators acting possibly in different Hilbert spaces. As some of the ``metrics'' do not fulfill all properties of a metric (e.g. some lack the triangle inequality or the definiteness), we call them ``distances'' here. To the best of our knowledge, such distances are treated for the first time here. The present article shall serve as a starting point of further research (see e.g. arXiv:2412.13165).
Paper Structure (13 sections, 15 theorems, 62 equations, 1 figure)

This paper contains 13 sections, 15 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Operator convergence in varying Hilbert spaces
  3. Metrics on sets of operators acting in different Hilbert spaces
  4. Related works
  5. Distances between operators acting in different Hilbert spaces
  6. A spectral distance for operators acting in different Hilbert spaces
  7. Quasi-unitary equivalence
  8. Consequences of quasi-unitary equivalence
  9. A distance arising from quasi-unitary equivalence
  10. Laplacians on thin branched manifolds shrinking towards a metric graph
  11. Metric graphs
  12. Thin branched manifolds ("fat graphs")
  13. Convergence of the Laplacian on thin branched manifolds

Key Result

Proposition 2.2

The function $d_{\text{\normalfont uni}}$ is a pseudometric on $\mathcal{B}_{(0,1]}$ (i.e., it is a metric execpt for the positive definiteness). Moreover, $d_{\text{\normalfont uni}}(R,\widetilde{R})=0$ is equivalent with the fact that there is a sequence of unitary operators $U_n \colon \mathscr H Finally, if $R$ and $\widetilde{R}$ are unitarily equivalent, then $d_{\text{\normalfont uni}}(R,\w

Figures (1)

  • Figure 1: A part of a metric graph, the (scaled) building blocks and the corresponding (part of a) thin branched manifold (here, $Y_e=\mathbb{S}^1$, and $X_\varepsilon$ is the surface of the pipeline network). The vertex neighbourhoods $X_{\varepsilon,v}$ are drawn in gray.

Theorems & Definitions (35)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4: herbst-nakamura:99
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 25 more