Explicit eigenvalue estimates for transfer operators

TL;DR

This paper derives explicit bounds for the eigenvalues of transfer operators acting on holomorphic function spaces, providing a clear quantitative understanding of their spectral decay.

Contribution

It offers explicit eigenvalue bounds for transfer operators on Bergman spaces, advancing spectral analysis in complex dynamical systems.

Findings

Eigenvalues decay exponentially with explicit bounds

Bounds depend on the dimension and properties of the operator

Provides tools for spectral estimates in complex analysis

Abstract

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if D is any open set in C^d, and L is a suitable transfer operator acting on Bergman space A^2(D), its eigenvalue sequence lambda_n(L) is bounded by |lambda_n(L)| \leq A\exp(-a n^{1/d}), where a, A are explicitly given.