This monograph deals with traces and determinants of pseudodifferential operators (\(\psi\)dos). More specially, the algebra of integer order classical \(\psi\)dos over a compact boundaryless manifold is endowed with a unique trace, the so-called residue trace. If one considers sufficiently small subalgebras, then new traces appear, as the classical trace on the subideal of smoothing operators. On the other hand, any functional extension of the classical trace to the full algebra of \(\psi\)dos is not tracial, but defines a quasi-trace instead; the residue trace arises in any such process as an obstruction to the traciality of the extension. The resulting residue-trace characters and quasi-trace characters provide an array of spectral geometric and topological invariants of the manifold.
For the sake of completeness we shall give two formulas illustrating the above introduction. A smoothing operator acting on the space of smooth sections of a vector bundle \(E\to M\) is a continuous linear operator \(A: C^\infty(M,E)\to C^\infty(M, E)\) such that
\[ (A\psi)(x)= \int_M k(x,y)\psi(y)\,dy,\tag{1} \]
where the Schwartz kernel \(k(x,y)\) is smooth on \(M\times M\). The unique (up to a scalar multiple) trace on the algebra of such operators is
\[ \text{Tr}(A)= \int_M \text{tr}(k(x,x)).\tag{2} \]
(2) is the classical trace, \(\text{Tr}(A)= \sum\mu\), i.e., \(\text{Tr}(A)\) is equal to the convergent sum over the discrete eigenvalues of the (compact) operator \(A\).
As for the quasi-trace of a \(\psi\)do \(A\) it can be defined via the regularized heat trace
\[ \text{TR}_{\text{quasi}}(A)= \lim_{t\to 0^+}\text{Tr}(e^{-t\Delta} A),\tag{3} \]
where \(\lim_{t\to 0^+}\) picks out the coefficient in front of \(t^0\) in the asymptotic expansion of \((e^{-t\Delta})\) for \(t\to 0^+\). As usual, \(e^{-t\Delta}\) is the heat operator, i.e., the smoothing operator which solves for \(t> 0\) the heat equation \(\partial_t f(x,t)+\Delta f(x,t)= 0\) on \(M\) with delta function initial data.
The book is divided into five chapters. Chapter 1 deals with general ideas of traces on algebras and with tracial structures of \(\psi\)dos. In Chapter 2 the abstract notion of determinant structures is developed. It is worth mentioning that many important invariants involve not only traces but also some logarithmic structures. The predominant part of the proofs and the detailed constructions concerning the \(\psi\)do traces and determinants are given in Chapter 4 “Pseudodifferential operator trace formulae”. Interesting results and considerations are exposed there on about 200 pages. Chapter 3 entitled “Computations, transition formulae, and the local index formula” contains rather precise computations of \(\psi\)do zeta traces and determinants. Moreover, the famous Arthur-Selberg trace formulae are also discussed. In the second part of Chapter 3 the author proposes transition formulae between resolvent trace asymptotics, heat trace asymptotics, and the singularity structure of spectral zeta functions and applies the above-mentioned results to relative determinant formulae. Section 3.4 is devoted to a residue trace computation, while in Section 3.5 the analytic (“local”) Atiyah-Singer index formula is proved. We do not give here any of the corresponding formulae because of the lack of space.
We point out that the residue trace is a trace, i.e., it vanishes on commutators, while quasi-traces are not. Any extension of the classical trace (3) in general does not vanish on commutators.
As for Chapter 5, entitled “Geometric families of pseudodifferential operators and determinant line bundles”, it deals with a more special problem, namely how the trace and determinant constructions described in the previous four chapters can be adapted to different families of \(\psi\)dos associated to geometric fibrations. As an application the following three examples are given: (1) conformal anomaly and the Polyakov formula, (2) Quillen’s computation of determinants of Cauchy-Riemann operators over a Riemann surface and (3) parametrix formulae.
Each chapter ends with a final section of notes. The author proposes there some additional comments on the proofs and references to the corresponding source materials. As for the references, they are given at the end of the monograph (pp. 664–676).
In conclusion the book of Simon Scott is rather interesting and well written. It contains a lot of results concerning different spectral geometric and topological invariants for classical \(\psi\)dos on smooth compact boundaryless manifolds. The referee recommends it to mathematicians and physicists working in \(\psi\)dos theory, spectral theory, complex analysis, geometry and topology, \(K\)-theory, quantum field theory etc.