Nuclearity and Grothendieck-Lidskii formula for quaternionic operators

TL;DR

This paper develops a trace concept for quaternionic operators, defines a quaternionic Fredholm determinant, and proves an analog of the Grothendieck-Lidskii formula, extending classical spectral results to quaternionic settings.

Contribution

It introduces a new trace notion for quaternionic operators, constructs the quaternionic Fredholm determinant, and generalizes the Grothendieck-Lidskii formula to quaternionic Fredholm operators.

Findings

Defined a trace for quaternionic operators using companion matrices

Established a quaternionic Fredholm determinant for trace-class operators

Proved an analog of the Grothendieck-Lidskii formula in quaternionic Hilbert spaces

Abstract

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace.