Permanence of invariant subspaces under functional calculus and related topics

In this study several similarity-invariant classes of operators on a complex separable Hilbert space are considered. All these classes are defined in terms of an operator and its square having the same properties(generating the same algebras, having the same invariant subspaces or the same commutant). The interiors and some of the closures of these classes(in the norm topology) are characterized in terms of the various parts of the spectrum. Certain new techniques to approach non commutative approximation problems are developed.