On the \(\Pi\)-operator in hyperholomorphic function theory
The authors study a generalization of the complex one-dimensional \(\Pi\)-operator in spaces of quaternionic-valued functions. The following main results could be obtained: 1. For Lipschitz continuous functions a representation formula is given in the case of general structure sets, which generalizes a former result of the reviewer. 2. The operator \({^{\phi,\psi}\Pi}\) is in \(L_2\) an isometry. 3. An orthogonal decomposition of the Hilbert space \(L_2\) in the case of general structure sets is given. 4. Two-sided invertibility of the \({^{\phi,\psi}\Pi}\)-operator is shown within the space \[ \text{im}^{\overline\phi}{\mathcal Q}\cap \text{im}^{\overline\psi}{\mathcal Q}, \] where \(\text{im}^{\overline\phi}{\mathcal Q}={^{\overline\phi}D}(\mathring W^1_2)\) and \({^{\overline\phi}D}\) is the Dirac operator. With this paper an essential progress in the development of the quaternionic operator calculus is done. Unfortunately, through the whole paper a very strange denotation for the quaternionic Lebesgue space \(L_p\) is used. This space is denoted by \(\text{Cl}_p\), which stands normalwise for a Clifford algebra over \(\mathbb{R}^p\) and can lead to misunderstandings.
scientific article; zbMATH DE number 1437573