Dixmier trace and Wodzicki residue

There is a third method of computing the logarithmic divergence of the spectrum of $\Dslash$, by means of residue calculus applied to powers of pseudodifferential operators. We shall give (only) a brief outline of what is involved.

Suppose that $H$ is an elliptic pseudodifferential operator on $\Ga(M, E)$ that extends to a positive selfadjoint operator (also denoted here by $H$) on the Hilbert space $L^2(M,E)$, which is defined as the completion of $\set{s \in \Ga(M,E) : \int_M \pairing{s}{s} \nu_g <\infty}$ in the norm $\Vert\psi\Vert := \sqrt{\braket{\psi}{\psi}}$, where

$$\braket{\phi}{\psi} := \int_M \pairing{\phi}{\psi} \nu_g$$

is the scalar product introduced in Section . We have in mind the example $H = \vert\Dslash\vert = (\Dslash^2)^{1/2}$ or else $H = (\Dslash^2 + 1)^{1/2}$, in case $\ker\Dslash \neq \{0\}$.

Since $M$ is compact, the operator $H$ on $L^2(M,E)$ is known to be Fredholm [Tay], thus $\ker H$ is finite dimensional. We can define its powers $H^{-s}$, for $s \in \bC$, by holomorphic functional calculus:

$$H^{-s} := \inv{2\pi i} \oint_\Ga \la^{-s} (\la - H)^{-1} d\la$$

where $\Ga$ is a contour that winds once anticlockwise around the spectrum of $H$, excluding $0$ to avoid the branch point of $\la^{-s}$. (We define $H^{-s}\psi := 0$ for $\psi \in \ker H$.)

By applying the same Cauchy integrals to the complete symbol of $H$, one can show that $H^{-s}$ is pseudodifferential, and obtain much information about its integral kernel. This was first done by Seeley [See]. He found that the following properties hold.

• If $H$ has order $d > 0$, then for $\Re s > n/d$, $H^{-s}$ is traceless and $\zeta_H(s) := \Tr H^{-s}$ is holomorphic on this open half-plane.
• For $x \neq y$, the function $s \mapsto K_{H^{-s}}(x,y)$ extends from the half-plane $\Re s > n/d$ to all of $\bC$, as an entire function.
• For $x = y$, the function $s\mapsto K_{H^{-s}}(x,x)$ can be continued to a meromorphic function on $\bC$, with possible poles only at $\set{s = (n - k)/d : k = 0,1,2,\dots}$.
• The residues at these poles are computed by integrating certain symbol terms over the sphere $\vert\xi\vert = 1$ in $T_x^* M$.

Later on, Wodzicki [Wodz] made a deep study of the spectral asymptotics of these operators, and in particular found that at $s = n/d$, the operator $H^{-n/d}$ is of order $(-n)$, and the residue at this pole depends only on its principal symbol; in fact,

$$\Res_{s=n/d} K_{H^{-s}}(x,x) \vert d^n x\vert = \frac{1}{d(2\pi)^n} \wres_x H^{-n/d}.$$

(For the proof, one applies Seeley's theory to $H = A^{-1}$.)

A basic result in noncommutative geometry is Connes' trace theorem of 1988 [Con1], which shows that this residue is actually a Dixmier trace.

We omit the proof, but a few comments can be made. In view of what was already said, it is enough to establish the first equality. The elliptic operator $H = A^{-1}$, of order $n$, has compact resolvent [Tay], so that $A$ itself is compact (we ignore any finite-dimensional kernel). If the eigenvalues of $A$ are $\la_k = s_k(A)$ (listed in decreasing order), the first equality reduces to the following known theorem on divergent series:

For a proof of the Proposition, see [GVF, pp. 294-295].

Next note that both $\Tr^+ A$ and $\Wres A$ are bilinear in $A$, so we can weaken the positivity hypothesis when comparing them. (There are other zeta-residue formulas available which are bilinear in $A$, but we do not go into that here.)