On the Characterization of the Intermediate Space in Generalized Factorizations

The study of systems of singular integral equations of CAUCHY type, of TOEPLITZ and WIENER -HOPF operators leads to the question of existence and representation of generalized factorizations of matrix functions Φ in L^P(Γ, σ)]^m. This yields a corresponding factorization of the basic multiplication or translation invariant operator A = A_CA₊, respectively, which can be seen as a splitting of a bounded into unbounded operators. The present paper is devoted to the study of the nature of the induced intermediate space Z = im A₊ =im A^?1, in particular, for Γ = ? and Φ ε ζC^β(?)]^{m × m} which is of special interest in certain applications. As we know, this implies detailed results about the structure and the explicit asymptotic behaviour of solutions of boundary and transmission problems near singular points with a relation also to eigenvalue problems which result from the classical series expansion approach or from the Mellin symbol calculus (see 13]).