On factorization of triangular matrices of the second order (Q1378070)

Let \(W\) be a rational matrix function and let \(\lambda_i\) be the poles of the entries of \(W\). If \(W\) has no poles on the real line and at infinity, then using the Laurent series expansion of \(W\) it is possible to find matrices \(A\), \(B\), \(C\), \(D\) such that \(W(\lambda)=D+C(\lambda-A)^{-1}B\), \(\lambda\in{\mathbb R}\). Also under certain assumptions there are projections \(P\), \(P^\times\), and \(\Pi\) such that \(W(\lambda)=W_{-}W_{+}\), where \(W_{-}=D+C(\lambda-A)^{-1}(I-\Pi)B\) and \(W_{+}=I+ D^{-1}C\Pi(\lambda-A)^{-1}B\). For the details see the book by \textit{I. Gohberg, S. Goldberg} and \textit{M. A. Kaashoek} [Classes of linear operators. Vol. II. (1993; Zbl 0789.47001)]. The author considers the \(2\times 2\) matrix \(W\) with entries \(a_{ij}\), where \(a_{11}=t^{-k}\), \(a_{12}=0\), \(a_{21}=\sum_{-k+1}^{k-1}a_jt^j\) (\(a_k\in{\mathbb C}\)), \(a_{22}= t^k\), and \(t=(\lambda-i)/(\lambda+i)\). Observing that there is some freedom in the choice of \(A\), \(B\), \(C\), \(D\) above, he is able to give for all these matrices the explicit form required in the two factors of \(W(\lambda)\).