Classification of ill-posedness for bounded linear operators in Banach spaces

TL;DR

This paper reviews and classifies different notions of well- and ill-posedness for linear operators in Banach spaces, highlighting differences from Hilbert spaces and discussing various types and borderline cases.

Contribution

It provides a comprehensive overview and classification of ill-posedness types for operators in Banach spaces, clarifying existing definitions and introducing new distinctions.

Findings

Different definitions of ill-posedness in Banach spaces are compared.

Classification of type I and type II ill-posedness is proposed.

Examples illustrate the distinctions and borderline cases.

Abstract

In this article, concepts of well- and ill-posedness for linear operators in Hilbert and Banach spaces are discussed. While these concepts are well understood in Hilbert spaces, this is not the case in Banach spaces, as there are several competing definitions, related to the occurrence of uncomplemented subspaces. We provide an overview of the various definitions and, based on this, discuss the classification of type I and type II ill-posedness in Banach spaces. Furthermore, a discussion of borderline (hybrid) cases in this classification is given together with several example instances of operators.