Every element b of a radical
Banach algebra ℛ satisfies limn→∞∥bn∥1∕n= 0. We are concerned here with the
existence of a lower bound for the rate of decrease of the sequence (∥bn∥)
under various assumptions over b and ℛ, when b is not nilpotent and ℛ
commutative.
If the nilpotents are dense in ℛ then for every sequence (λn) of positive reals
there exists a nonnilpotent b ∈ℛ such that liminfn→∞∥bn∥∕λn= 0. A stronger
result holds if ℛ possesses furthermore a bounded approximate identity. On the other
direction if ℛ has no nilpotent element and if some element of ℛ which is not a
divisor of zero acts compactly on ℛ then there exists a sequence (λn) of positive reals
such that liminfn→∞∥bn∥∕λn= +∞ for every nonzero b ∈ℛ. Also there
exists universal lower bounds for the rate of decrease of ∥an∥ if (at) is an
analytic semigroup over the positive reals or over some open angle. Such lower
bounds do not exist for infinitely differentiable semigroups over the positive
reals.
