Explicit almost-periodic factorization for a class of triangular matrix functions
Let \(APW\) be the Banach algebra of almost periodic series \[ f=\sum_j c_j e^{ix\lambda}\quad\text{with}\;\;\lambda_j\in\mathbb{R},\;c_j\in\mathbb{C}\;\;\text{and}\;\;\|f\|_W=\sum_j |c_j|<\infty, \] and let \(APW^\pm\) be the Banach subalgebras of \(APW\) constituted by functions with \(\pm\lambda_j\geq 0\), respectively. An almost periodic matrix function \(G\) admits a (right) canonical \(APW\) factorization if it is represented in the form \(G=G_-G_+\), where the entries of the matrix functions \(G_\pm, G_\pm^{-1}\) are in \(APW^\pm\), respectively. Under conditions of existence, a canonical \(APW\) factorization, which automatically is a canonical generalized factorization relative to \(L_2(\mathbb{R})\), is constructed explicitly for triangular almost periodic matrix function of the form \[ G=\begin{bmatrix} E^{-1}&0\\ a E^\alpha+b+cE^{-\beta}&E\end{bmatrix}, \] where \(E(\xi)=e^{i\xi}\) for \(\xi\in\mathbb{R}\), \(\alpha,\beta\in(0,1)\), \(\alpha/\beta\notin\mathbb{Q}\), \(a,b,c\in\mathbb{C}\setminus\{0\}\), and either (i) \(n\alpha+\beta=1\) for some \(n\in\mathbb{N}\) and \(\beta>(n-1)\alpha\), or (ii) \(2\alpha+\beta=1\) and \(\beta<\alpha\). In case (i), the matrix function \(G\) admits a canonical \(APW\) factorization if and only if \(|c/b||a/b|^{\beta/\alpha}\neq 1/n\). In case (ii), such a factorization is constructed under the following sufficient conditions: either \(|c/b||a/b|^{\beta/\alpha}<1/2\), or \(|c/b||a/b|^{\beta/\alpha}>m/(m+1)\), where \(m=\big[\frac{\alpha}{\beta}\big]+1\). The existence of almost periodic polynomial solutions is also studied.
scientific article; zbMATH DE number 5279121
scientific article