Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

✍ Scribed by Bernhard Gramsch; Winfried Kaballo

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 873 KB
• Volume: 204
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19992040106

✦ Synopsis

In this article we construct multiplicative decompoeitions of holomorphic Fkedholm operator d u d functions on Stsin manifolds with d u e s in d o u a algebras of differential and pseudo differential operahm which are submultiplicrtive 0' -algebras, a concept introduced by the first author. For Redholm functions T ( z ) sathtjing M obvious topological condition we. prove (0.1)

T ( z ) = A(z)(I + S(r)) ,

where A(%) is holomorphic and invertible and S(z) is holomorphic with values in an "arbitrarily small" operator ideal. This is a stronger condition on S(r) than in the authors' additive dccompoeition theorem for meromorphic inversee of holomorphic hadholm functions [12], where the "mnallneas" of S ( r ) depends on the number of complex variables. The Multiplicative Decompwition theorem (0.1) sharpens the authors' Regularization theorem [ll]; in crate of the B a n d algebra L ( X ) of all bounded linear operators on a B a n d space, (0.1) has been proved by J. LEfieReff 1201 for one complex variable and by M. 0. ZAZDENBERQ, S. 0. K m , P. A. KUCHYWT m d A. A. PANKOV (261 for the B a n d ideal of compact operators.