Perturbation properties of the generalized spectral radius

Let n∈N, K∈{R,C} and Mn(K) be the set of all n-by-n matrices with entries in K. We investigate the sensitivity of the generalized spectral radius ρK(A):=max{|λ|:λ∈Kand|Ax|=|λx|foranx∈Kn∖{0}}of A∈Mn(K) under perturbations, i.e. we ask how ρK(A) is related to ρK(A+E) for perturbations E∈Mn(K). For example and somewhat surprisingly, Elsner's famous bound on the spectral variation of two complex square matrices fully translates to |ρC(A)−ρC(B)|⩽∥A+B∥1−1n2∥A−B∥1n2for all A,B∈Mn(C), where ∥⋅∥2 is the matrix 2-norm. For ρR this result holds true locally.