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Functional calculus for Safarov pseudo-differential operators

Santiago Gómez Cobos, Michael Ruzhansky

TL;DR

This work develops a holomorphic functional calculus for Safarov-type global pseudo-differential operators on closed manifolds with a (not necessarily metric) connection. Using parameter-ellipticity, it constructs resolvent parametrices and proves that f(A) defined by a Dunford–Riesz integral remains within the same global symbol class, with a computable asymptotic symbol expansion. The approach yields concrete results for complex powers, exponentials, and logarithms of operators, and enables Szegö-type determinant formulas, heat-kernel trace expansions, and spectral ζ-functions, thus providing a robust toolkit for global spectral invariants. Overall, the paper extends Seeley's holomorphic calculus to Safarov's broader operator classes and establishes powerful trace and spectral tools for geometric analysis.

Abstract

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $ζ$-functions.

Functional calculus for Safarov pseudo-differential operators

TL;DR

This work develops a holomorphic functional calculus for Safarov-type global pseudo-differential operators on closed manifolds with a (not necessarily metric) connection. Using parameter-ellipticity, it constructs resolvent parametrices and proves that f(A) defined by a Dunford–Riesz integral remains within the same global symbol class, with a computable asymptotic symbol expansion. The approach yields concrete results for complex powers, exponentials, and logarithms of operators, and enables Szegö-type determinant formulas, heat-kernel trace expansions, and spectral ζ-functions, thus providing a robust toolkit for global spectral invariants. Overall, the paper extends Seeley's holomorphic calculus to Safarov's broader operator classes and establishes powerful trace and spectral tools for geometric analysis.

Abstract

Given a smooth, closed Riemannian manifold equipped with a linear connection (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral -functions.

Paper Structure

This paper contains 16 sections, 25 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some geometric generalities
  4. Safarov calculus
  5. Bessel potentials
  6. A trace on L
  7. Resolvent of pseudo-differential operators
  8. Parameter-ellipticity
  9. Norm estimates of resolvent operators
  10. Norm estimates of resolvent operators
  11. Holomorphic functional calculus
  12. Examples
  13. Applications
  14. Szegö limit theorem
  15. Traces of heat operators
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian manifold and let $\nabla$ be a linear connection (not necessarily metric). Let $0\leq \delta<\rho \leq 1$, $m\geq 0$, $A\in \Psi_{\rho, \delta}^{m}(\Omega^{\kappa}, \nabla, \tau)$, let $\Lambda\subset {\Bbb C}$ be a sector and let $f$ be a holomorphic function on $ Suppose that $\sigma_A$, the symbol of $A$, is parameter-elliptic with respect to $\Lambda$. Moreo

Theorems & Definitions (65)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 55 more