Some results on spectra and certain norms

Given the norms of powers ( xⁿ)n≥ 0 of a Banach algebra element x, the largest possible value of the minimum modulus on the spectrum of x is determined. It is also shown that, given a Banach algebra element x and a compact set K with maximum modulus no more than the spectral radius of x, there exists a Banach algebra element y with yⁿ= xⁿ for all n≥ 0 and spectrum equal to the union of the spectrum of x and K. These results, along with the spectral radius formula, are generalized to the joint spectrum of several commutative Banach algebra elements. The generalization of the spectral radius formula presented gives the maximum possible joint spectrum for commutative Banach algebra elements x₁,…,xₙ, given the norms ( x₁i₁⋯ xₙiₙ)i₁,…,iₙ≥ 0.