Some results on non-self-adjoint operators, a survey

TL;DR

This survey reviews recent advances in understanding non-self-adjoint operators, focusing on spectral properties, asymptotics, and pseudo-spectrum, highlighting new results in these complex spectral problems.

Contribution

The paper compiles and discusses recent findings on spectral asymptotics and pseudo-spectrum for non-self-adjoint operators, providing a comprehensive overview of current research developments.

Findings

Analysis of Kramers-Fokker-Planck operators

Spectral asymptotics in two dimensions

Weyl asymptotics for eigenvalues with small random perturbations

Abstract

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems.