This research note presents a unified analysis of the results obtained in this area of operatory theory. The author is concerned with (norm) approximation problems related with subsets of the algebra of all operators acting on a complex separable infinite dimensional Hilbert space that are invariant under similarity. Many natural families fall into this category, and the author presents a set of general techniques to deal with problems which include: closure of a similarity-invariant set of operators and, more generally, distance to that set, fine structure of polynomially compact operators, compact perturbations and approximation, inner derivations with closed range, points of spectral continuity and the structure of the closure of a similarity orbit. The book should be of interest to researchers and postgraduate students in functional analysis, operator theory and certain areas of quantum mechanics. The second volume by C.Apostol, L.A.Fialkow, D.A.Herrero and D.Voiculescu completes the results contained here.
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