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On the pseudospectra of Berezin-Toeplitz operators

David Borthwick, Alejandro Uribe

TL;DR

The paper develops a semiclassical microlocal framework for Berezin-Toeplitz operators to analyze pseudospectra of non-selfadjoint operators. By coupling polarized Hermite distributions with Hörmander subellipticity, it constructs phase-space localized pseudomodes when the Poisson bracket of the symbol’s real and imaginary parts is negative, and proves resolvent lower bounds when it is positive, while also determining the asymptotic behavior of the numerical range. The authors illustrate the theory with explicit quantizations of the torus and the complex projective line, providing concrete matrix models and localization results for pseudomodes. A key contribution is the demonstration that the numerical range converges to the convex hull of the symbol image and the establishment of a non-selfadjoint weak Szegő limit theorem for Berezin-Toeplitz operators, with significant implications for spectral stability in semiclassical settings.

Abstract

We estimate the norm of the resolvent of non-selfadjoint Berezin Toeplitz operators in the semi-classical limit, under various assumptions on the Poisson bracket of the real and imaginary parts of the symbol. In case this bracket is negative, we symbolically construct pseudomodes well-localized in phase space. We also show that the numerical range converges to the convex hull of the image of the symbol. Our techniques involve the theory of Hermite distributions and classical subelliptic results of Hörmander.

On the pseudospectra of Berezin-Toeplitz operators

TL;DR

The paper develops a semiclassical microlocal framework for Berezin-Toeplitz operators to analyze pseudospectra of non-selfadjoint operators. By coupling polarized Hermite distributions with Hörmander subellipticity, it constructs phase-space localized pseudomodes when the Poisson bracket of the symbol’s real and imaginary parts is negative, and proves resolvent lower bounds when it is positive, while also determining the asymptotic behavior of the numerical range. The authors illustrate the theory with explicit quantizations of the torus and the complex projective line, providing concrete matrix models and localization results for pseudomodes. A key contribution is the demonstration that the numerical range converges to the convex hull of the symbol image and the establishment of a non-selfadjoint weak Szegő limit theorem for Berezin-Toeplitz operators, with significant implications for spectral stability in semiclassical settings.

Abstract

We estimate the norm of the resolvent of non-selfadjoint Berezin Toeplitz operators in the semi-classical limit, under various assumptions on the Poisson bracket of the real and imaginary parts of the symbol. In case this bracket is negative, we symbolically construct pseudomodes well-localized in phase space. We also show that the numerical range converges to the convex hull of the image of the symbol. Our techniques involve the theory of Hermite distributions and classical subelliptic results of Hörmander.

Paper Structure

This paper contains 20 sections, 27 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Preliminaries
  4. Setup and strategy
  5. Polarized Hermite distributions
  6. Proofs
  7. Proof of Theorem \ref{['I']}
  8. Proof of Theorem \ref{['II']}
  9. Examples
  10. A preliminary example
  11. Quantization of the torus
  12. The Hilbert spaces
  13. Matrix coefficients
  14. Localization of pseudomodes
  15. The complex projective line
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $T_f=\{T^{(N)}\,,\,N=1,2,\ldots\}$ be a Berezin-Toeplitz operator with smooth principal symbol $f:X\to{\mathbb C}$. 0. For all $\lambda\in{\mathbb C}$, 1. Assume that $\lambda = f(x_0)$ where $x_0\in X$ is such that Then there exists a sequence of vectors $\{\psi_N\in{\mathcal{H}}_N\}$ with microsupport precisely $\{x_0\}$ and such that 2. On the other hand, if $\lambda = f(x_0)$ and th

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Lemma 3.1
  • ...and 39 more