Double Operator Integration with applications to Quantised Calculus

Abstract

The theory of double operator integration provides a powerful set of tools for the study of spectral asymptotics of compact operators. We give a self-contained overview of the theory from its foundations, including a complete proof of the fundamental Peller's theorem. The theory is developed with the goal of proving a formula for the difference of complex powers of self-adjoint operators, which has recently been applied to problems in Connes' quantised calculus. The final two chapters give applications to the Conformal Trace Theorem for the Hausdorff measure of Julia sets of quadratic polynomials and to the characterisation of quantum differentiability on noncommutative Euclidean spaces.