Traces and commutators

The following is a true (and well-known) theorem: $$\newcommand{\Tr}{\mathop{\rm Tr}}$$

Suppose $$A$$ and $$B$$ are bounded operators on a Hilbert space, and $$AB$$ and $$BA$$ are trace class.  Then $$\Tr(AB)=\Tr(BA)$$.

This is easy to prove if one of the operators $$A,B$$ is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works).  In the general case it is a bit harder.  The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of $$AB$$ and $$BA$$ are the same (including multiplicities).  Continue reading

Proofs and Understanding

A year and a half ago I wrote a post on my ideas about using “structured proofs” to improve understanding in the Analysis I course.

I duly tried this approach, and felt that it made some difference, though perhaps not as much as I had hoped.

Now in this month’s Notices of the AMS there is a fascinating article by a team of educators from Loughborough University in the UK.  Entitled Understanding and Improving Undergraduate Proof Comprehension, the article discusses as three-stage effort to help undergraduate students gain a deeper understanding of proofs – the same issue that my “structured proof” software was intended to address.

A correction to EOTAM

In my book Elliptic operators, topology and asymptotic methods (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981.  I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press).  Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link.  In this post I want to explain what is garbled and how the useful part of the argument can be recovered.  Continue reading

A “well known interpolation formula”

In the Atiyah-Bott paper on their Lefschetz theorem for elliptic complexes, they give a very nice elementary example of the Lefschetz theorem for the Dolbeault complex, by considering the automorphism of  ${\mathbb C}{\mathbb P}^n$ given by

$[z_0,\ldots, z_n] \mapsto [\gamma_0z_0, \ldots, \gamma_nz_n ]$

in homogeneous coordinates, where the $$\gamma_i$$ are distinct and nonzero complex numbers.  This has $$(n+1)$$ simple fixed points and applying the holomorphic Lefschetz theorem gives

$1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}.$

This is Example 2 on p. 460 of the second Atiyah-Bott paper.  They go on to describe this as a “well known interpolation formula”. Continue reading

Artin’s Criterion, Part II

In this post I want to sketch the proof of “Artin’s criterion”, following Ahlfors’ book as referred to in my previous post (and presumably following Artin himself, though I still have not come up with any original reference to him).  The argument generalizes easily to $$(n-1)$$-dimensional cycles in an open subset $$\Omega\subseteq {\mathbb R}^n$$, but for simplicity I will stick to the planar case.