Trace-Class and Nuclear Operators

TL;DR

This paper traces the development of nuclear and trace-class operators from Banach space tensor products to Hilbert space operators, explaining their equivalence and underlying concepts.

Contribution

It clarifies the relationship between nuclear and trace-class operators, showing their equivalence in the context of Hilbert spaces.

Findings

Nuclear and trace-class operators are shown to be equivalent in Hilbert spaces.

The paper connects tensor product theory with operator classes.

It provides a historical and conceptual overview of these operator classes.

Abstract

This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.