About explicit factorization of some classes of non-rational matrix functions. (English) Zbl 1131.47016
Continuing a series of papers on the factorisation properties of operator-valued matrices, the authors consider the class of matrix-valued functions
\[
A_\gamma(b)=\begin{pmatrix} 1 & b \\ \overline b & | b| ^2 + \gamma \end{pmatrix} ,
\]
where \(b\in L_\infty(\mathbb T)\), \(\gamma\in\mathbb C\setminus\{0\}\) and \(\mathbb T\) is the unit disc in \(\mathbb C\). It is shown that the existence of a canonical generalized factorisation of \(A_\gamma(b)\) is equivalent to the fact that \(\gamma\) lies in the resolvent set of a certain operator depending on \(b\). Assuming that such a factorisation exists, the authors give an algorithm how it can be constructed. In the last section, this algorithm is illustrated by an example.
Reviewer: Monika Winklmeier (Bern)
MSC:
| 47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
| 47A10 | Spectrum, resolvent |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
Keywords:
explicit factorization; factorization algorithm; singular integral operator; inner function; outer functionReferences:
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