TOPOLOGICAL DIVISOR OF ZERO PERTURBATION FUNCTIONS

Let ( ) ⋅ , A be a complex unital Banach algebra and let Eq N be the set of all algebra-norms on A equivalent to the given algebra-norm. In this paper, we introduce the concept of d ρ perturbation and rd ρ perturbation functions depending on a norm Eq N ∈ ⋅ and related to the notion of “topological divisors of zero”. We prove that some usual measures of either non-compactness or nonstrict-singularity of operators, as well other quantities are d ρ perturbation or rd ρ perturbation function. We prove several spectral radius formulae for d ρ perturbation and rd ρ perturbation functions. In particular, we prove that if P ⋅ is a d ρ perturbation or rd ρ perturbation function and , A ∈ x then