Index concepts for linear differential-algebraic equations in finite and infinite dimensions

TL;DR

This paper compares various index concepts for linear differential-algebraic equations in Banach spaces, highlighting their differences in finite and infinite dimensions through theoretical analysis and examples.

Contribution

It introduces and compares multiple index concepts in Banach spaces, clarifying their relationships and differences in finite and infinite-dimensional settings.

Findings

All indices are equivalent in finite dimensions.

In infinite dimensions, indices are generally not equivalent.

Examples illustrate the complex relationships among indices.

Abstract

Different index concepts for linear differential-algebraic equations are defined in the general Banach space setting, and compared. For regular finite-dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite-dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.