Constructing non-compact operators into $c_0$

TL;DR

The paper demonstrates that for any dense non-compact operator between Banach spaces, there exists a composition with a map into c_0 that remains non-compact, extending the Josefson-Nissenzweig Theorem.

Contribution

It generalizes the Josefson-Nissenzweig Theorem by constructing non-compact operators into c_0 from dense non-compact operators.

Findings

Existence of a linear operator T:Y→c_0 such that TS:X→c_0 is not compact

Generalization of the Josefson-Nissenzweig Theorem

Applicable to all dense non-compact operators between Banach spaces

Abstract

We prove that for each dense non-compact linear operator $S:X\to Y$ between Banach spaces there is a linear operator $T:Y\to c_0$ such that the operator $TS:X\to c_0$ is not compact. This generalizes the Josefson-Nissenzweig Theorem.