Decomposition of a scalar operator into a product of unitary operators with two points in spectrum
Let \(U^\alpha\) be the set of unitary operators on a separable Hilbert space \(H\) whose spectra lie in \(\{1,e^{i\alpha}\}\). Let \(\Omega_k^\alpha\) be the set of all unitary \(u\in \mathbb{C}\) for which a solution of \(uI=U_1 U_2 \cdots U_k,\,\,\,U_j\in U^\alpha\) exists. The authors show that \(\Omega_k^\alpha \supset \{e^{i\alpha x} | x \in [1+\frac{1}{k-3},\,k-1-\frac{1}{k-3}]\}\) for \(k>4\). This is one of the main results in this paper. The authors mention the equation \(e^{i(A_1+A_2+\cdots +A_k)}=e^{i{\tilde{A}_1}}e^{i{\tilde{A}_2}} \cdots e^{i{\tilde{A}_k}}\), where \(\tilde{A}_j\) is a matrix unitarily similar to \(A_j\). They investigate topics that are related to this equation.
scientific article; zbMATH DE number 5775694
scientific article