Bernhard Gramsch
Oka's principle and the division of distributions for analytic Fredholm functions
The complex analytic homotopy theory for Fredholm operators depending analytically or continuously on parameters is applied to the division of vector valued distributions. This new connection is also based on some classical contributions (e.g. Math.Ann. 214 , 95 - 147; 1975) and some work with W. Kaballo (Math.Nachr. 204, 83 - 100; 1999). The Oka - Grauert - Gromov principle is discussed for special Fréchet manifolds of semi Fredholm operators and idempotent elements in algebras of pseudodifferential operators. As source spaces for the Oka principle we can admit in some cases holomorphy regions in DFN - spaces with basis. The Hörmander class $(1,1)$ is known to be not spectrally invariant in any $\mathcal{L}(H)$, where $H$ is a Hilbert space. But some amount of a holomorphic Fredholm theory can be derived also in this case using lifting methods. A series of problems is mentioned for the operator valued Oka - principle in connection with submanifolds in Fréchet algebras.