A factorization theorem for analytic functions operating in a Banach algebra

Philip C. Curtis, Jr. and Henrik Stetkaer

Vol. 37 (1971), No. 2, 337–343

DOI: 10.2140/pjm.1971.37.337

Abstract

Cohen’s factorization-theorem asserts that if the Banach algebra A has a left approximate identity, then each y ∈ A may be written y = xz,x,z ∈ A. The vector x may be chosen to be bounded by some fixed constant and z may be chosen arbitrarily close to y. In this setting the theorem below asserts that if F is a holomorphic function defined on a sufficiently large disc about ζ = 1, and satisfying F(1) = 1, then each y ∈ A may be written y = F(x)z, where 𝜃j,Z ∈ A. Again x may be chosen to be bounded by some fixed constant and z may be chosen close to y.

Mathematical Subject Classification 2000

Primary: 46H05

Secondary: 46K05

Milestones

Received: 10 October 1970

Published: 1 May 1971