Functional calculus and joint torsion of pairs of almost commuting operators

TL;DR

This paper explores the behavior of determinants and joint torsion for pairs of almost commuting Fredholm operators, analyzing their functional calculus and connections to algebraic symbols, with applications to operator theory.

Contribution

It introduces new variational formulas for joint torsion and extends results on determinants and Tate symbols within the framework of Fredholm modules.

Findings

Derived variational formulas for joint torsion.

Extended results on determinants and Tate symbols.

Analyzed the commutation properties of functional calculus with projections.

Abstract

This paper investigates the transformation of determinants of pairs of Fredholm operators with trace class commutators. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we recover in particular some results of R. Carey and J. Pincus on determinants and Tate tame symbols. Additionally, we obtain variational formulas for joint torsion.