Abstract
Let L(H) be the algebra of operators on a complex separable Hilbert space H, let T ε L(H) and let ℓ(T) = {WTW1 : W is invertible in L(H)} be the “ similarity orbit ” of T. The norm closurey ℓ(N)− of ℓ(N) is completely characterized “up to compact perturbations” for N normal. A partial answer is also given for normal operators in non-separable Hilbert spaces. For the separable case, it is shown that if the spectrum of T is the union of a totally disconnected set and finitely many pairwise disjoint subsets of regular analytic Jordan arcs (or Jordan curves) and is perfect, then ℓ(T)− = ℓ(A)− for every A having the same spectrum as T. If T is a spectral operator, then ℓ(T)− contains the “scalar part” of T. The results also include some information about those operators T such that ℓ(T)− is maximal or minimal with respect to inclusion, in the family of the sets {ℓ(A)− : A ɛ L(H)}.
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