2023/06/25 by Paula Cerejeiras, Cerejeiras, Paula, Fabrizio Colombo +7 · 1 citation
Mathematics · Physics and Astronomy · #46S05 #47S05 (Primary) 47B10 (Secondary) #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2306.14189
openalex publication_date 2023/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 (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.