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On Generalized Strong and Norm Resolvent Convergence

Gerald Teschl, Yifei Wang, Bing Xie, Zhe Zhou

TL;DR

This work develops a streamlined framework for generalized strong and norm resolvent convergence of self-adjoint operators defined on varying Hilbert spaces via embedding operators $J_n$. It proves that generalized norm resolvent convergence yields convergence of $J_n^* f(A_n) J_n$ to $f(A)$ for bounded $f$, and that generalized strong resolvent convergence yields the corresponding strong limit, with robust implications for spectra and spectral projections. The authors provide practical criteria based on relative boundedness or quadratic-form bounds to verify convergence and demonstrate the approach with Sturm–Liouville operators with varying weights, establishing conditions for generalized norm resolvent convergence and essential spectrum stability under asymptotic changes of coefficients. The results generalize and streamline prior approaches, removing the need for projection embeddings and semiboundedness, and have implications for approximation of singular Sturm–Liouville problems and related PDE operators on graphs and weighted spaces.

Abstract

We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral projections. In addition, we give some applications to Sturm-Liouville operators.

On Generalized Strong and Norm Resolvent Convergence

TL;DR

This work develops a streamlined framework for generalized strong and norm resolvent convergence of self-adjoint operators defined on varying Hilbert spaces via embedding operators . It proves that generalized norm resolvent convergence yields convergence of to for bounded , and that generalized strong resolvent convergence yields the corresponding strong limit, with robust implications for spectra and spectral projections. The authors provide practical criteria based on relative boundedness or quadratic-form bounds to verify convergence and demonstrate the approach with Sturm–Liouville operators with varying weights, establishing conditions for generalized norm resolvent convergence and essential spectrum stability under asymptotic changes of coefficients. The results generalize and streamline prior approaches, removing the need for projection embeddings and semiboundedness, and have implications for approximation of singular Sturm–Liouville problems and related PDE operators on graphs and weighted spaces.

Abstract

We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral projections. In addition, we give some applications to Sturm-Liouville operators.
Paper Structure (3 sections, 17 theorems, 61 equations)

This paper contains 3 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Generalized strong and norm resolvent convergence
  3. Applications to Sturm--Liouville operators

Key Result

Theorem 2.3

Let $A_n$, $A$ be self-adjoint operators and suppose that $A_n$ converges to $A$ in generalized norm resolvent sense. Then for every bounded continuous function $f\!:\! \Sigma\! \to \! {\mathbb C}$ with $\lim\limits_{\lambda\to-\infty}f(\lambda)=\lim\limits_{\lambda\to\infty}f(\lambda)$. If $A_n$ converges to $A$ in generalized strong resolvent sense, then for every bounded continuous function $

Theorems & Definitions (34)

  • Example 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 24 more