Factorizability of Matrix Functions: a Direct Proof

The factorization theorem mentioned in the title is about matrix-valued functions satisfying the Lipschitz condition on the real line and is related to the Riemann problem. Many questions of mathematical physics reduce to what is called the Riemann problem, which consists in finding a certain scalaror vector-valued piecewise holomorphic function. This function is assumed to be regular inside and outside of some contour, and its boundary values on that contour must exist and satisfy some special matching conditions. Solving a vector Riemann problem reduces to the factorization of matrix-valued functions. Provided the contour is finite, the proof of factorizability under natural assumptions dates back to Plemelj (1908); see [5]. A considerable contribution to this range of ideas was made in the books [6, 7]; among more recent publications we mention [8]. For many problems it is essential that the contour on which the matrix to be factorized is given coincide with the real axis. For an important and general class of such matrices, factorizability was proved by Gokhberg and Krĕın in their classical paper [9]. That proof is indirect. The factorization problem for an (n × n)-matrix E + G(t) (E is the unit matrix) is equivalent to finding linearly independent solutions of the homogeneous vector integral equation (∗) h(t) + ∫ +∞ 0 G̃(t− s)h(s) ds = 0, s, t ≥ 0, h ∈ C, where G̃(t) is the matrix whose Fourier image coincides with G(t). Integral equations similar to (∗) were the starting point of investigations in [9]. Besides the theory of integral equations, in the constructions of [9] an important part was played by Wiener’s subtle results pertaining to harmonic analysis. In the present paper, the author’s aim is to prove the factorizability of matrix-valued functions of t ∈ (−∞,+∞) along the lines suggested by Plemelj. This proof is direct. The matrices to be factored belong to the class Lipα (0 < α < 1), which is fairly standard in problems of mathematical physics. The fact that this class is natural was justified by many authors for finite contours; the same is done in the present paper for the contour R = (−∞,+∞). Professor A. P. Kachalov turned my attention to the fact that, with the help of an appropriate conformal transformation, the factorization problem in question can be reduced to the case of a finite contour. This makes it possible to use the existing results for finite contours and to simplify some constructions. However, we consider integral equations in the form in which they arise, with R in the role of the integration contour. It may be expected that these integral equations can be used for a numerical solution of the problem. Compared with the finite contour, the case of R has some specifics. A series of “additional theorems” have turned out to be necessary; hopefully, these results are of some interest in themselves. 2000 Mathematics Subject Classification. Primary 30E25. Supported by RFBR (grant no. 08-01-00511). c ©2008 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use