The factorization problem—A survey

The problem of factoring a positive Hermitian operator into the product of a causal operator and its adjoint is reviewed. Three classes of factorization are studied, miniphase factorization, regular factorization, and special factorzation. The former always exists, and a complete representation thereof is obtained in terms of the properties of reproducing kernel spaces. The regular factorization does not exist, in general, and its existence theroy is shown to be equivalent to the unitary equivalence problem of classical operator theory. Finally, an existence and uniqueness theorem for special factors is formulated in terms of certain Cauchy integrals and is shown to be closely related to Wiener-Hopf theory.