Spectral theory of infinite dimensional dissipative Hamiltonian systems

TL;DR

This paper develops a spectral theory for infinite-dimensional dissipative Hamiltonian systems, focusing on singular operator pencils and their impact on solution uniqueness, revealing differences from finite-dimensional cases.

Contribution

It introduces and analyzes three concepts of singularity for operator pencils in infinite dimensions, linking them to solution uniqueness in dissipative differential-algebraic equations.

Findings

Classical finite-dimensional results do not always extend to infinite dimensions.

Singularity of operator pencils is crucial for solution uniqueness in dissipative systems.

Examples illustrate subtle differences between concepts of singularity.

Abstract

The spectral theory for operator pencils and operator differential-algebraic equations is studied. Special focus is laid on singular operator pencils and three different concepts of singularity of operator pencils are introduced. The concepts are analyzed in detail and examples are presented that illustrate the subtle differences. It is investigated how these concepts are related to uniqueness of the underlying algebraic-differential operator equation, showing that, in general, classical results known from the finite dimensional case of matrix pencils and differential-algebraic equations do not prevail. The results are then studied in the setting of structured operator pencils arising in dissipative differential-algebraic equations. Here, unlike to the general infinite-dimensional case, the uniqueness of solutions to dissipative differential-algebraic operator equations is closely related to the singularity of the pencil.