Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions
The authors investigate matrices whose entries are almost periodic functions in various senses (Bohr, Besicovitch or with absolutely convergent Fourier series). By \((AP)^{n\times n}\) one denotes the algebra of square matrices with (complex) almost periodic entries, of type \(n\) by \(n\). A matrix function \(G(t)\) in \((AP)^{n\times n}\) is said to possess a factorization of \(G(t)= G_+(t) G_-(t)\), \(t\in\mathbb{R}\), with \(G_+\) and \(G_-\) invertible for each \(t\in\mathbb{R}\), both almost periodic with their inverses, and such that the spectra of \(G_+\) and \((G_+)^{-1}\) are in \(\mathbb{R}_+\), while the spectra of \(G_-\) and \((G_-)^{-1}\) are in \(\mathbb{R}_-= (-\infty,0]\). A marathon type theorem is given, providing 16 equivalent conditions for the factorization of a matrix function \(G(t)\). Some of these conditions are, in fact, invertibility conditions for \(G(t)\) or some attached operators. It is shown, among other properties of factorizable matrices with almost periodic entries, that once the factorization holds with almost periodic matrices \(G_+\) and \(G_-\), it also holds when requiring the absolute convergence of the Fourier series of the factor matrices. The proof of the theorem requires several nontrivial auxiliary results related to function algebras. Besides various particular cases, the basic theorem is used in connection with the Sarason interpolation concept and the Toeplitz corona theorem.
scientific article; zbMATH DE number 1267973
scientific article