Generalized canonical factorization of matrix and operator functions with definite Hermitian part (Q1198114)

Let \(W(\lambda)\) be a regular rational (square) matrix function (which may have poles on \(\mathbb{R}\) or at \(\infty\)); \(W(\lambda)\) is said to have a left canonical pseudo-spectral factorization if \(W(\lambda)=W_ +(\lambda) W_ -(\lambda)\), where \(W_ +(\lambda)\) (resp. \(W_ - (\lambda)\)) is a rational matrix function without poles or zeros (a zero of a regular rational matrix function \(W(\lambda)\) is, loosely speaking, a zero of \(\text{det }W(\lambda)\)) in the open upper (resp. lower) halfplane, and at every point of \(\mathbb{R}\) and at \(\infty\) the factorization is minimal (i.e., there are no pole-zero cancellations between \(W_ +\) and \(W_ -\) on \(\mathbb{R}\cup\infty\)). The authors prove the existence of such a factorization (and of its right counterpart) under the assumption that the Hermitian part \[ W_ H(\lambda)= {\textstyle{1\over 2}} (W(\lambda)+ W(\overline{\lambda})^*) \] of \(W\) has no zeros on \(\mathbb{R}\cup\infty\) and that \(W_ H(\lambda)\) is positive definite for each \(\lambda\in\mathbb{R}\) which is not a pole of \(W_ H\). They provide formulae for the factors in case \(W\) has neither poles nor zeros on \(\mathbb{R}\), and the polynomial case is considered as well; further, the uniqueness and the stability of the factorization are studied. Finally, they extend part of the results to the infinite dimensional case, assuming \(W(\lambda)\) to be a regular meromorphic rational operator function (loosely speaking, this means that \(W(\lambda_ 0)\) is invertible for some \(\lambda_ 0\) and for each point \(\lambda_ 1\in\mathbb{C}\) there exists a closed subspace \(M\) of finite codimension such that \(W(\lambda)|_ M\) has a Fredholm-valued continuation on a full neighbourhood of \(\lambda_ 1\)).