Norms of Powers in Banach Algebras

Abstract

By showing the existence of certain functions in A+ (the algebra of analytic functions in the unit disc with absolutely convergent Taylor series), we prove that if T is a power bounded operator on a Banach space X, and x ∈ X satisfies Tx ≠ x, then ∑ n − 1 ∞ ‖ ( 1 − T ) n x ‖ ‖ ( 1 − T ) n − 1 x ‖ diverges .