A note on the Goldberg-Thorp example in light of the classification of linear ill-posed problems in Banach spaces

TL;DR

This paper examines a specific hybrid-type linear operator in Banach spaces, analyzing its structure, properties, and implications for regularization and stability in ill-posed problems.

Contribution

It provides a detailed analysis of the Goldberg-Thorp example within the classification of ill-posed operators, highlighting its structural properties and regularization challenges.

Findings

The operator $B$ is a strictly singular, non-closed range operator from $\, ext{ell}^1$ to $ ext{ell}^2$.

Null-spaces of hybrid-type operators are not complemented, affecting stability and regularization.

The paper summarizes the properties of $B$ and discusses limitations of regularization methods.

Abstract

This note considers the strictly singular mapping, denoted by $B$, from $\ell^1$ onto $\ell^2$ of an example by Goldberg and Thorp from 1963 as a typical hybrid-type operator in the context of the classification of ill-posed linear operators in infinite-dimensional Banach spaces. The null-spaces of hybrid-type operators are not complemented and therefore need special attention. More generally, a given well-posedness definition for linear operators requiring both closed range and complemented null-space is motivated by the continuity of occurring pseudo-inverse operators as a stability criterion. With respect to the operator $B$, structure, representation and properties of the operator and its adjoint are summarized in a theorem. Moreover, limitations and opportunities of regularization approaches for the treatment of $B$ are outlined.