Preface to the Second Edition In 1977, I gave some lectures at the University of Texas which described the general theory of trace ideals initiated by von Neumann and Schatten (Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979). Because this theory has many different kinds of applications, the lecture notes I produced at the time were widely used, and I got many requests for information on how to obtain it once it fell out of print. In 1993, I lectured at a summer school in Vancouver on the theory of ap- plications of rank one perturbations of self-adjoint operators (Spectral analysis of rank one perturbations and applications in: Mathematical Quantum Theory. II. Schrodinger Operators, CRM Proc. Lecture Notes, 8, pp. 109-149, American Math- ematical Society, Providence, RI, 1995). The two topics have much in common. My interest in each arose in my research in several problems at once. And, of course, rank one perturbations are an extreme case of compact perturbations. Thus, when I started exploring the possibility of reprinting the Trace Ideals book as a second edition, it was natural to combine it with my Vancouver lectures. In preparing this new edition, I had to decide first whether to totally rewrite the material, and I chose not to because the basic theory hasn't changed much. Once I made that decision, I felt it made sense to only lightly edit the material, so, for example, references to theorems or equation numbers would be the same. I fixed typos and made a few references to the addendum, especially where a conjecture had been settled. But I followed the original texts closely so much so that the notation in Chapters 1-10 (the original Texas lectures) and Chapters 11-14 (the Vancouver lectures) are, in a few points, slightly different. I did add a much better index and an addendum describing some developments since the original notes were written. It is a pleasure to thank many people who helped on this project: R. Bing, J. Dollard, and J. Gilbert for the invitation to give the original lectures in Texas and J. Feldman, R. Froese, and L. Rosen for the invitation to lecture in Vancouver. V. Jaksic made useful remarks about the addendum. The proofreading was done while I was a Lady Davis Visiting Professor at the Hebrew University of Jerusalem, and I am grateful for the hospitality of the Mathematics Department there provided by H. Farkas and Y. Last. Because of the form of the original notes, it was necessary to TeX them, a task performed admirably by Cherie Galvez. Finally, I acknowledge Martha's love, which makes it all easier. Barry Simon Jerusalem, Israel February 2005

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