The K-theory and the invertibility of almost periodic Toeplitz operators
Let \(\Gamma\) be an additive subgroup of \({\mathbb{R}}\). Let \(AP(\Gamma)\) be the set of almost periodic functions \(\phi\) on \({\mathbb{R}}\) with the Fourier series \(\phi (t)=\sum_{\lambda \in \Gamma}a_{\lambda}e^{2\pi i\lambda t}\). Let \(H^ 2({\mathbb{R}})\) be the Hardy space of the upper half- plane and let \(P:L^ 2({\mathbb{R}})\to H^ 2({\mathbb{R}})\) be the orthogonal projection. For \(\phi\in AP(\Gamma)\) the Toeplitz operators are defined as \(T_{\phi}=PM_{\phi}| H^ 2({\mathbb{R}})\). It is well-known that when \(\Gamma\) is dense in \({\mathbb{R}}\), \(T_{\phi}\) is invertible if and only if \(\phi\) is invertible and the topological index of \(\phi\) is 0. Using the K-theory the author studies the invertibility of the Toeplitz operator \(T_{\Phi}\) on \(H^ 2({\mathbb{R}})\otimes {\mathbb{C}}^ n\) with a matrix symbol \(\Phi \in AP(\Gamma)\otimes M_ n\), where \(M_ n\) is the set of \(n\times n\) matrices.
scientific article; zbMATH DE number 4050347
scientific article