Pseudodifferential Operator Trace Formulae
In this chapter details of the construction of the fundamental trace functionals on pseudodifferential operators are given. The approach to traces taken here is through an analysis of the singularity structure of the operator Schwartz kernel. This allows for a dual perspective on traces, viewed either from the microanalytic approach of pseudodifferential methods or from the approach favoured in applications in geometric analysis, differential geometry, and theoretical physics, of subtracting-off the singular part of the kernel. Regularized traces arise via subtracting-off from the Schwartz kernel the meromorphic continuation of homogeneous distributions defined by the pseudodifferential operator symbol, linking trace regularization methods with traditional distributional analysis. Exact formulae are computed for regularized trace functionals for log-classical pseudodifferential operators, allowing, in particular, precise formulae for the zeta determinant. The final part of the chapter analyses the principal multiplicative functional on the semiqroup of pseudodifferential operators.
Keywords: Distributions, homogeneous, Schwartz kernel, singularity structure, extended trace, residue trace, log-classical, log polyhomogeneous symbols, residue determinant, proofs, construction quasi-trace, trace defect formulae
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