Multiplicative decompositions of holomorphic Fredholm functions and Psi*-algebras

In this article we construct multiplicative decompositions of holomorphic F redholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submu ltiplicative psi* -algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we prov e (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an "arbitrarily small" operator ideal. This is a stronger condition on S(z) than in the authors' additive decompos ition theorem for meromorphic inverses of holomorphic Fredholm functions [1 2], where the "smallness" of S(z) depends on the number of complex variable s. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Reg ularization theorem [11]; in case of the Banach algebra L(X) of all bounded linear operators on a Banach space, (0.1) has been proved by J. LEITERER [ 20] for one complex variable and by M. G. ZAIDENBERG, S. G. KREIN, P. A. KU CHMENT and A. A. PANKOV [26] for the Banach ideal of compact operators.