On invariant subspaces for rational Toeplitz operators (Q1895335)

Let \(H^2 (\mathbb{T})\) be the Hardy space of the unit circle \(\mathbb{T}\), \(P\) the orthogonal projection of \(L^2 (\mathbb{T})\) to \(H^2 (\mathbb{T})\), and \(T_\varphi (f)= P(\varphi f)\), \(\forall f\in H^2 (\mathbb{T})\) the Toeplitz operator, where \(\varphi\in L^\infty (\mathbb{T})\). In this paper, the author constructs a nontrivial invariant subspace of \(T_\varphi\), when \(\varphi\) is a nonconstant rational function with no poles in \(\mathbb{T}\) such that \(\varphi (\mathbb{T})= \sigma (T_\varphi)\). The method is based on a contour integral, with a contour intersecting the spectrum \(\sigma (T_\varphi)\) at two points.