The general theme of the course is the spectrum of operators on Hilbert space, with an emphasis on explicit calculation of spectra (and of spectrally related things like traces, when appropriate). After a review of essential information about Hilbert space and bounded operators on Hilbert space, we will treat compact operators, especially the selfadjoint ones. Many interesting examples of compact operators with more or less computable spectrum and trace come from inverting unbounded operators associated to ODE and PDE problems, so we will have a look at unbounded operators. The situation of operators with entirely continuous spectrum will be represented by the calculation of the spectrum of the combinatorial Laplacian of a homogeneous tree, and by other examples from infinite combinatorics and elsewhere as time permits.
(Reihani 2011 )