Introduction

Summary

These chapters contain the material of a summer course (8 weeks) given at LSU a few years ago and repeated at Johannes Kepler Universität, Linz. In the summer the mathematics department at LSU is faced with offering courses that may be taken by graduate students at all levels: beginning to advanced Ph.D.! I hope that this material meets (and fills) that need.

For these lectures, the student will need a bit of mathematical sophistication and a fairly good course in advanced calculus (Cauchy Sequences, convergence of sequences, uniform continuity) and a good course in (finite dimensional) linear algebra (determinants, eigenvalues, linear transformations).

An undergraduate course in complex variables would also be nice. But, if the student was introduced to the line integral in calculus, the complex integration we do in these notes should present no difficulties. Knowledge of Lebesgue measure is not assumed. Thus, these notes will not discuss, e.g. L₂[0,1] and thus also will not discuss integral operators given by L₂-kernels. (To the student: Forget this paragraph if it fails to make sense.)

Many will say that this omits too much from the theory of compact operators on Hilbert space. I claim not. It omits many important examples but in these notes we are interested in the representation of compact operators. From this point of view we have omitted nothing!

Our goal is to prove the Lidskij trace formula in as easy a fashion as possible.