Cohen's Factorization Method Using an Algebra of Analytic Functions

Introduction The motivating example for the main theorem of this paper is the following elementary factorization in the Banach algebra c0 of complex sequences converging to zero. If a; is in c0, then there is an a ^ 0 in c0 such that x is in f(a).c0 for each continuous function / on [0, oo) with J-i{0} = {0} and/not tending to zero 'too quickly' as t tends to zero. This observation is essentially a restatement of the trivial observation that for each sequence tending to zero there is a sequence tending to zero more slowly. We also observe that the element a has a functional calculus stronger than the analytic functional calculus. In this paper a generalization of this result is proved. The algebra c0 is replaced by a Banach algebra with a bounded approximate identity bounded by 1, and the functions that operate on a are taken to be analytic and bounded in the open disc A = {zeC:|z — £ | < £} and to converge to zero as | z | tends to zero faster than some \z\° (a > 0). This technical factorization theorem, which is the main result of this paper, is stated near the end of the introduction. The motivating example given above was clearly behind Salem's factorization L{T) ^ ( T ) = ^(T), where T is the circle group [17]. The proof of our main result uses the basic idea introduced by Cohen in [6] that has been used in all factorizations in general Banach algebras A with bounded approximate identities (see [12, 7, 3, 18, 10]). An identity is adjoined to A and a sequence of invertible elements bn is constructed in A © C1 so that bn converges to an element in A and bn~ .x converges even though bn~ x is an unbounded sequence in A. The exact definition of the bn and estimates used vary in the different papers but this stage of the proof is the same. Several authors have taken suitable analytic functions of the bn to obtain factorizations stronger than x = ay (see [7, 3, 10, 18], and [2], which is special case of [7]). If / is analytic on A, then we shall define f(bn) using the analytic functional calculus and show that this sequence is convergent for suitable / and suitable choice of the bn. The weights used in our definition of bn are the terms of a divergent geometric series, whereas in Cohen's proof the weights are from a convergent geometric series [6].