Vol. 94, No. 1, 1981

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Rates of decrease of sequences of powers in commutative radical Banach algebras

Jean Esterle

Vol. 94 (1981), No. 1, 61–82

Every element b of a radical Banach algebra satisfies limn→∞bn1∕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 , when b is not nilpotent and commutative.

If the nilpotents are dense in then for every sequence (λn) of positive reals there exists a nonnilpotent b ∈ℛ such that liminf n→∞bn∕λn = 0. A stronger result holds if possesses furthermore a bounded approximate identity. On the other direction if has no nilpotent element and if some element of which is not a divisor of zero acts compactly on then there exists a sequence (λn) of positive reals such that liminf n→∞bn∕λn = +for every nonzero b ∈ℛ. Also there exists universal lower bounds for the rate of decrease of anif (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.

Mathematical Subject Classification 2000
Primary: 46J05
Received: 25 June 1979
Revised: 14 May 1980
Published: 1 May 1981
Jean Esterle