Reducing and minimal reducing subspaces of slant Toeplitz operators (Q1989139)

Let \(\mathbb T\) be the unit circle in the complex plane and let \(e_{n}(z)=z^{n}\), \(n\in\mathbb Z\), \(z\in\mathbb T\), be the orthonormal basis in the space \(L^{2}(\mathbb T)\). For a function \(\varphi\in L^{\infty}(\mathbb T)\), the slant Toeplitz operator \(A_{\varphi}\) on the space \(L^{2}(\mathbb T)\) is defined by the equality \(A_{\varphi}=W\varphi I\), where \(W\) is the linear continuous operator on the space \(L^{2}(\mathbb T)\) defined by the relation \(W(e_{2n})=e_{n}\), \(W(e_{2n-1})=0\), \(n\in\mathbb Z\). For the operators \(A_{\varphi}\) in the case \(\varphi(z)=z^{N}\), \(N\in\mathbb Z\), there is found an infinite set of minimal reducing subspaces. There are investigated some properties of reducing subspaces of these operators. We mention here one of these results. Let \(N_{1}\) and \(N_{2}\) be the integers with odd difference, \(\varphi(z)=z_{N_{1}}\), \(\psi(z)=z^{N_{2}}\), \(z\in\mathbb T\). Then any minimal reducing space of \(A_{\varphi}\) cannot be reducing for \(A_{\psi}\).