Factorization of a class of matrix functions in the Wiener algebra of order 2 (Q6047508)

The Wiener algebra \(W\) on the real line \(\mathbb R\) by definition consists of functions \(c+{\mathcal F}f\), where \(c\in\mathbb C\), \(f\in L_1(\mathbb R)\), and \(\mathcal F\) stands for the Fourier transform. In turn, \(W_\pm\) are the subalgebras of \(W\) corresponding to \(f\) supported in \(\mathbb R_\pm :=\{x\in\mathbb R\colon \pm x\geq 0\}\). It is a classical result by \textit{I. C. Gochberg} and \textit{M. G. Krejn} [Usp. Mat. Nauk 13, No. 2(80), 3--72 (1958; Zbl 0098.07402)] that any invertible \(n\)-by-\(n\) matrix function \(G\) with the entries in \(W\) admits a factorization \(G_+DG_-\). Here \(G_\pm\) and their inverses are matrices with the entries in \(W_\pm\) while \[ D(x)=\text{diag }\left[ \left(\frac{x-i}{x+i}\right)^{\kappa_1},\ldots, \left(\frac{x-i}{x+i}\right)^{\kappa_n} \right]. \] The integers \(\kappa_1,\ldots,\kappa_n\) are defined by \(G\) uniquely and are called the partial indices of \(G\). Computing the factorization, and the partial indices in particular, is a challenging problem. The main result of the paper under review is that for \[ G(x)=\begin{bmatrix} 1 & e^{ix\tau_0}m^-(x) \\ e^{-ix\tau_0}m^+(x) & 1+m^+(x)m^-(x)\end{bmatrix} \] where \(\tau_0>0\) and \(m^\pm\in W_\pm\), the factorization is canonical (i.e., \(\kappa_1=\kappa_2=0\)) provided that \(||m^-||_W\cdot ||m^+||_W<1/4\). This result is used to obtain a solvability criterion for the convolution type equation on a finite interval: \[ u(t)-\int_0^\tau k(t-s)u(s)\, ds= f(t), \quad t\in (0,\tau). \]