A simple proof of the Fredholm Alternative

TL;DR

This paper provides a straightforward proof of the Fredholm Alternative for certain classes of operators, including compact and pseudodifferential operators, enhancing understanding of their spectral properties.

Contribution

It introduces a simplified proof of the Fredholm Alternative for compact operators approximated by finite rank operators and extends the result to pseudodifferential operators of order less than zero.

Findings

Proof of the Fredholm Alternative for compact operators as norm limits of finite rank operators

Extension of the Fredholm Alternative to pseudodifferential operators of order < 0

Simplification of existing proofs for better accessibility

Abstract

In this expository note, we present a simple proof of the Fredholm Alternative for compact operators that are norm limits of finite rank operators. We also prove a Fredholm Alternative for pseudodifferential operators of order < 0.