The Criterion of Factorability. Ф-Factorization and its Basic Properties

Abstract

In this chapter we shall study relationships between properties of the Riemann boundary value problem for a piecewise analytic vector function and the factorability of its coefficient, i.e. the matrix function G. It is well-known that in the classical case, if we seek for a solution of the Riemann problem with a Hölder matrix G, the factorability of G is equivalent to the Fredholmness of the boundary value problem. The transition to the solution of the Riemann problem with measurable matrix function G in the Smirnov classes E^± _p leads to a new quality. It turns out that the factorability of G, generally speaking, does not imply the Fredholmness of the corresponding boundary value problem. In fact, the vector-valued Riemann boundary value problem with a factorable matrix function G and the associate problem considered in the classes L_p and L_q, respectively, have finite defect numbers, and the indices of the problems are opposite. However, these problems are, in general, not normally solvable, i.e. their images can be not closed. Nevertheless, these images are in a sense well-situated. More strictly speaking, the following weakened closedness condition is fulfilled: the image of the Riemann boundary value problem (and of the problem associate to it) contains all those rational vector functions belonging to its closure. Moreover, the factorability of the matrix coefficient of a Riemann problem in L_p is equivalent to the property described above, a property intermediate between finiteness of the defect numbers and Fredholmness.