Factorization algorithm for some special non-rational matrix functions. (English) Zbl 1208.47017
Ball, Joseph A. (ed.) et al., Topics in operator theory. Volume 1: Operators, matrices and analytic functions. Proceedings of the 19th international workshop on operator theory and applications (IWOTA), College of William and Mary, Williamsburg, VA, USA, July 22–26, 2008. A tribute to Israel Gohberg on the occasion of his 80th birthday. Basel: Birkhäuser (ISBN 978-3-0346-0157-3/hbk; 978-3-0346-0163-4/set; 978-3-0346-0158-0/ebook). Operator Theory: Advances and Applications 202, 87-109 (2010).
This paper reviews several factorization results and their connections with other problems in analysis. The authors consider matrix-valued functions of the form
\[ A_\gamma(b)=\begin{pmatrix} e & b \\ b^* & b^*b+\gamma e \end{pmatrix}, \]
where \(e\) is the identity \(n\times n\) matrix, \(b\) is a matrix valued function with entries in \(L_\infty(\mathbb{T})\), and \(0\neq \gamma\in \mathbb{C}\). An algorithm is presented, which detects if such a matrix function \(A_\gamma(b)\) admits a generalized factorization; a \(k\times k\) matrix \(A\) admits a generalized factorization if
\[ A=A_+ \Lambda A_-, \]
where \(A_+\) and \(A_-\) are \(k\times k\) matrices with entries in (respectively) \(L_2^+(\mathbb T)\) and \(L_2^-(\mathbb T)\), \(\Lambda\) is a diagonal matrix with \(t^{j_1},\dots, t^{j_k}\) in the diagonal (\(j_1\geq\dots \geq j_k\) are integers), and \(A_+P_+A_+^{-1}\) and \(A_-P_+A_-^{-1}\) are bounded operators in \(L_2(\mathbb T)^n\). The algorithm also provides such factorizations.
For the entire collection see [Zbl 1181.47002].
\[ A_\gamma(b)=\begin{pmatrix} e & b \\ b^* & b^*b+\gamma e \end{pmatrix}, \]
where \(e\) is the identity \(n\times n\) matrix, \(b\) is a matrix valued function with entries in \(L_\infty(\mathbb{T})\), and \(0\neq \gamma\in \mathbb{C}\). An algorithm is presented, which detects if such a matrix function \(A_\gamma(b)\) admits a generalized factorization; a \(k\times k\) matrix \(A\) admits a generalized factorization if
\[ A=A_+ \Lambda A_-, \]
where \(A_+\) and \(A_-\) are \(k\times k\) matrices with entries in (respectively) \(L_2^+(\mathbb T)\) and \(L_2^-(\mathbb T)\), \(\Lambda\) is a diagonal matrix with \(t^{j_1},\dots, t^{j_k}\) in the diagonal (\(j_1\geq\dots \geq j_k\) are integers), and \(A_+P_+A_+^{-1}\) and \(A_-P_+A_-^{-1}\) are bounded operators in \(L_2(\mathbb T)^n\). The algorithm also provides such factorizations.
For the entire collection see [Zbl 1181.47002].
Reviewer: Esteban Andruchow (Los Polvorines)
MSC:
| 47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
| 47A10 | Spectrum, resolvent |
