Fréchet operator algebras with spectral invariance
by
Bernhard Gramsch
Gutenberg University, Mainz, Germany

The analytic perturbation theory for \Psi^*-algebras in the microlocal analysis covering C^\infty-structures made a decisive progress during the nineties. See e.g. the work of W. Kaballo, F. Mantlik, E. Schrohe, and the author in the Math. Nachr. 99, and of R. Lauter in J. Funct. Anal. 99. An essential tool for the complex analysis of operator functions on finite and infinite dimensional holomorphy regions \Omega with values in Fréchet algebras with an open group of invertible elements is the submultiplicativity. This has been proved recently for the Fréchet algebra of classical pseudo-differential operators with asymptotic expansions. This opens the possibility of many applications in a theory of inversion of stochastic partial differential equations, where the differential operators depend on infinite dimensional analytic parameters defined by stochastic data. On holomorphy regions in some DFS-spaces the nonabelian cohomology theory of Grauert-Cartan-Bungart-Leiterer with its many applications in operator theory is derived in the spirit of the Oka principle. This has direct consequences for the analytic theory of stochastic PDE's. The results seem to be new even for functions on \Omega with values in homogeneous spaces of Fredholm operators of a Hilbert space.

(T)

Date received: April 14, 2000