Abstract
This thesis is concerned first with a non-compact variation of Connes'
trace theorem, which demonstrated that the Dixmier trace extends the
notion of Lebesgue integration on a compact manifold. To obtain the
variation, we develop a new $\zeta$-residue formula, which is proved by an
innovative approach using double operator integrals. Using this formula,
Connes' trace theorem is shown for operators of the form $M_f(1-\Delta)^{-
\frac{d}{2}}$ on $L_2(\mathbb{R}^d)$, where $M_f$ is multiplication by a
function belonging to the Sobolev space $W_1^d(\mathbb{R}^d)$---the space
of all integrable functions on $\mathbb{R}^d$ whose weak derivatives up to
order $d$ are all also integrable---and $\Delta$ is the Laplacian on
$L_2(\mathbb{R}^d)$. An analogous formula for the Moyal plane is also
shown.
The $\zeta$-residue formula we derive also enables a second result. We
consider the smoothed Riesz map $\mathrm{g}$ of the massless Dirac
operator $\mathcal{D}$ on $\mathbb{R}^d$, for $d\geq 2$, and study its
properties in terms of weak Schatten classes. Our sharp estimates, which
are optimal in the scale of weak Schatten classes, show that the decay of
singular values of $\mathrm{g}(\mathcal{D}+V)-\mathrm{g}(\mathcal{D})$
differs dramatically for the case when the perturbation $V$ is a purely
electric potential and the case when $V$ is a magnetic one. The
application of double operator integrals also yields a similar result for
the operator $f(\mathcal{D}+V)-f(\mathcal{D})$ for an arbitrary monotone
function $f$ on $\mathbb{R}$ whose derivative is Schwartz.