Generalised factorisation for a class of Jones form matrix functions

Abstract

SynopsisA systematic approach is proposed for the generalised factorisation of certain non-rational n × n matrix functions. The first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means. It closes a gap between the classical Wiener-Hopf procedure and the operator theoretic method of generalised factorisation. Secondly, as examples we consider certain matrix functions of Jones form or of N-part form, which are equivalent to each other, in a sense. The factorisation procedure is complete and explicit, based only on the factorisation of scalar functions, of rational matrix functions and upon linear algebra. Applications in elastodynamic diffraction theory are treated in detail and in a most effective way.

One sided invertibility of matrices over commutative rings, corona problems, and Toeplitz operators with matrix symbols

“…Following Theorem 3.4 in [9] or Lemma 2.1 in [23], we see that there exist R ∈ GR n×n and M It follows from (3.4) and (3.8) that (3.3) holds, with F = G + .…”

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

“…where the factors M − and M + do not satisfy the conditions for (2.1) to be a Wiener-Hopf factorization of G. Of course, if we transform that meromorphic factorization into a Wiener-Hopf one (by use of the method presented in [8] or [16]) we get that information explicitly through the diagonal middle factor D.…”

“…We study here a class of matrix functions of the so called Jones form [8,14], i.e, of the form where a j ∈ L ∞ (R) and R ∈ R n×n satisfies the relation R n = sI n with s ∈ GR (I n denoting the identity matrix in C n×n ). For such matrix functions R we can write…”

“…In this section we study the relation between two types of factorization of matrix functions, meromorphic and Wiener-Hopf factorizations, from a point of view different from that of [8] and [16], leading to a procedure to turn the first into the second which ultimately consists in solving a linear system of equations. We start with the following.…”

“…This kind of factorization of matrix symbols does not give, as least directly, the same amount of information on Toeplitz operators as the Wiener-Hopf factorization defined above, namely as regards invertibility. However, a systematic method to turn a meromorphic factorization into a WH-factorization was presented in [16] and [8] involving only a finite number of elementary algebraic operations and rational factorization. As it is remarked in [8], in a certain sense, this closed the gap between the classical and the operator-theoretic approaches to Wiener-Hopf problems.…”