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Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry

Siarhei Finski

TL;DR

The paper addresses the problem of understanding small eigenvalues of Toeplitz operators on polarized complex projective manifolds and their logarithmic distribution through Mabuchi geometry and Lebesgue envelopes. It develops a transfer-operator framework showing that ($\frac{1}{k}$) times the logarithm of a Toeplitz operator is asymptotically Toeplitz with symbol $\phi(h^L, NZ)$, and it connects the spectral data to Mabuchi geodesic speed; under Lebesgue pluriregularity, stronger asymptotics are obtained. A central theme is the role of non-negligible psh envelopes $h^L_\mu$ and Lebesgue envelopes $h^L_{\rm Leb,A}$, which yield precise exponential decay rates for small eigenvalues and clarify when these envelopes coincide with classical psh envelopes. The work also develops Bernstein-Markov-type properties for Lebesgue measures, introduces an approximability framework for measures, and extends the theory to generalized Toeplitz operators and Toeplitz matrices, including explicit results for the circle case. Overall, the results provide a robust geometric-analytic mechanism linking semiclassical Toeplitz spectra to Mabuchi geometry, Lebesgue-enveloped metrics, and transfer-operator techniques, with implications for both complex geometry and numerical Toeplitz matrix analysis.

Abstract

We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices.

Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry

TL;DR

The paper addresses the problem of understanding small eigenvalues of Toeplitz operators on polarized complex projective manifolds and their logarithmic distribution through Mabuchi geometry and Lebesgue envelopes. It develops a transfer-operator framework showing that () times the logarithm of a Toeplitz operator is asymptotically Toeplitz with symbol , and it connects the spectral data to Mabuchi geodesic speed; under Lebesgue pluriregularity, stronger asymptotics are obtained. A central theme is the role of non-negligible psh envelopes and Lebesgue envelopes , which yield precise exponential decay rates for small eigenvalues and clarify when these envelopes coincide with classical psh envelopes. The work also develops Bernstein-Markov-type properties for Lebesgue measures, introduces an approximability framework for measures, and extends the theory to generalized Toeplitz operators and Toeplitz matrices, including explicit results for the circle case. Overall, the results provide a robust geometric-analytic mechanism linking semiclassical Toeplitz spectra to Mabuchi geometry, Lebesgue-enveloped metrics, and transfer-operator techniques, with implications for both complex geometry and numerical Toeplitz matrix analysis.

Abstract

We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices.

Paper Structure

This paper contains 12 sections, 39 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. Acknowledgement.
  4. Logarithm of a Toeplitz operator is asymptotically Toeplitz
  5. Determining measures and non-negligible psh envelopes
  6. Lebesgue dense points and psh envelopes
  7. Bernstein-Markov type properties for the Lebesgue measures
  8. From Toeplitz to Transfer operators and back
  9. Asymptotic class of the weighted $L^2$-norms
  10. Mabuchi geometry and the contact set
  11. Growth of balls of holomorphic sections and Lebesgue envelopes
  12. Generalized Toeplitz operators and Toeplitz matrices

Key Result

Theorem 1.1

Let $f \in L^{\infty}(X)$, $f \neq 0$, be such that $(K, h^L)$ is Lebesgue pluriregular and $\lambda(Z) = 0$. We let $c(f) := \max_{x \in X} \log(h^L(x) / h^L_{{\rm{Leb}}, K}(x))$. Then for any $\epsilon > 0$, there is $k_0 \in \mathbb{N}$, such that for any $k \geq k_0$, we have

Theorems & Definitions (86)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • ...and 76 more