When does ‖f(A)‖ = f(‖A‖) hold true?
Florian Bünger, Siegfried M. Rump · Linear and Multilinear Algebra · 2021
Let a strongly stable norm ∥⋅∥ on the set Mn of complex n-by-n matrices be given, which means that ∥Ak∥≤∥A∥k for all A∈Mn and all k=1,2,…. Furthermore, let f(x)=∑k=0∞ckxk be a power series with nonnegative coefficients ck≥0 and radius of convergence R>0. If ∥I∥>1, we additionally suppose that c0=f(0)=0. We aim to characterize those A with ∥A∥ 0 for all k≥1.