Wiener-Hopf factorization and factorization indices

Abstract

This chapter concerns canonical as well as non-canonical Wiener-Hopf factorization of an operator-valued function which is analytic on a Cauchy contour. Such an operator function is given by a realization with a possibly infinite dimensional Banach space as state space, and with a bounded state operator and with bounded input-output operators. The first main result is a generalization to operator-valued functions of the canonical factorization theorem for rational matrix functions presented earlier in Section 3.1. In terms of the given realization, necessary and sufficient conditions are also presented in order that the operator function involved admits a (possibly non-canonical) Wiener-Hopf factorization. The corresponding factorization indices are described in terms of certain spectral invariants which are defined in terms of the realization but do only depend on the operator function and not on the particular choice of the realization. The analysis of these spectral invariants is one of the main themes of this chapter.