On operators with many spectral components

For a complex Banach algebra A with unit and n∈ N, let Cₙ( A) be the set of all elements whose spectrum has at least n connected components, and let d_σ( A)=;\; Cₙ( A)≠\ be the spectral depth of A. We characterize nonemptiness of Cₙ( A) in terms of orthogonal idempotents, compute d_σ( A) for finite-dimensional algebras, and show that every class Cₙ( A) is open. For operators on an infinite-dimensional complex Banach space X these classes are dense in B(X), which implies that C_∞( B(X)) is a dense G_δ-subset of B(X). By contrast, we exhibit a commutative Banach algebra whose spectral classes are all nonempty and none of which is dense. For self-adjoint operators on a Hilbert space, we obtain sharper perturbation results, with positive semi-definite perturbations of prescribed rank producing spectra with exactly n components. We show that a quasinilpotent weighted forward shift Sw on X admits a nilpotent rank-one operator F such that the spectrum of Sw+αF has infinitely many components for every nonzero $α$.