On the spectral radius formula in Banach algebras
Jan-Erik Björk · Pacific Journal of Mathematics · 1972
B will always denote a commutative semi-simple Banach algebra with a unit element.If fe B then r(/) denotes its spectral radius.A sequence F=(/, )Γ is called a spectral null sequence if \\fj\\ ^ 1 for each j, while limy-*,r(fj) = 0.If F=(fj) is a spectral null sequence we put TN(F)limsupj-c \\ff \\ ιlN for each N S 1. Finally we define the complex number TN(B) -sup {r^CF): F is a spectral null sequence in B).In general rχ(B) = 1 for all N ^ 1 and the aim of this paper is to study the case when TN(B) < 1 for some N.