Geometry of the spectral semidistance in Banach algebras

Abstract

Let A be a unital Banach algebra over ℂ, and suppose that the nonzero spectral values of a and b ∈ A are discrete sets which cluster at 0 ∈ ℂ, if anywhere. We develop a plane geometric formula for the spectral semidistance of a and b which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that a and b are quasinilpotent equivalent if and only if all the Riesz projections, p(α, a) and p(α, b), correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space X in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu’s geometric formula (which requires the knowledge of the local spectra of the operators at each 0 ≠ x ∈ X). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results.