This is a second edition of a wellknown book stemming from the author's lectures on the theory of trace ideals in the algebra of operators in a Hilbert space. Because of the theory's many different applications, the book was widely used and much in demand. For this edition, the author has added four chapters on the closely related theory of rank one perturbations of selfadjoint operators. He has also included a comprehensive index and an addendum describing some developments since the original notes were published. This book continues to be a vital source of information for those interested in the theory of trace ideals and in its applications to various areas of mathematical physics. Readership Graduate students and research mathematicians interested in the theory of operators and its applications to mathematical physics. Reviews "...very well suited for a graduate course since the exposition is clear and perfectly selfcontained."  Mathematical Reviews "From a review of the first edition: "Beautifully written and well organized ... indispensable for those interested in certain areas of mathematical physics ... for the expert and beginner alike. The author deserves to be congratulated both for his work in unifying a subject and for showing workers in the field new directions for future development."  Zentralblatt MATH Table of Contents  Preliminaries
 Calkin's theory of operator ideals and symmetrically normed ideals; Convergence theorems for \(\mathcal J_P\)
 Trace, determinant, and Lidskii's theorem
 \(f(x)g(i\nabla)\)
 Fredholm theory
 Scattering with a trace condition
 Bound state problems
 Lots of inequalities
 Regularized determinants and renormalization in quantum field theory
 An introduction to the theory on a Banach space
 Borel transforms, the Krein spectral shift, and all that
 Spectral theory of rank one perturbations
 Localization in the Anderson model following AizenmanMolchanov
 The Xi function
 Addenda
 Bibliography
 Index
