Cyclic nearly invariant subspaces for semigroups of isometries (Q6564152)

Let \(S\) denote the unilateral forward shift on the Hardy space \(H^2(\mathbb{D})\). A closed subspace \(\mathcal{M}\subseteq H^2(\mathbb{D})\) is called nearly \(S^{*}\)-invariant if \(S^{*}f\) belongs to \(\mathcal{M}\) whenever \(f\in\mathcal{M}\) and \(f(0)= 0\). \textit{D. Hitt} [Pac. J. Math. 134, No. 1, 101--120 (1988; Zbl 0662.30035)] obtained a characterization of all nearly \(S^{*}\)-invariant subspaces. For a general left invertible bounded operator \(T\) on a Hilbert space \(\mathcal{H}\), a closed subspace \(\mathcal{M}\subseteq\mathcal{H}\) is called nearly \(T^{-1}\)-invariant if whenever \(g\in\mathcal{H}\) such that \(Tg\in\mathcal{M}\), it holds that \(g\in\mathcal{M}\). In the case \(T=S\), we see that \(S^{*}\)-invariant subspaces are precisely \(S^{-1}\)-invariant subspaces.\N\NThe results in the paper under review address the following two interesting problems.\N\N(1) Obtaining Hitt-like formula for the nearly \(T^{-1}_{\psi}\)-invariant subspaces, where \(\psi\) is an automorphism of the disc.\N\N(2) Describing the nearly \(T^{-1}_{\theta}\)-invariant subspaces, where \(\theta\) is a infinite Blaschke product.\N\NThe authors obtain a complete result for (1), using composition operators and model spaces. Additionally, they also describe the structure of the nearly invariant subspaces for discrete semigroups generated by (finitely or infinitely) many operators of the form \(T^{-1}_{\psi}\), where each \(\psi\) is an automorphism of the disc. On the other hand, (2) is closely related to the structure of nearly invariant subspaces of shift semigroups. The authors characterize nearly invariant cyclic subspaces generated by a rational outer function or a product of a rational outer function with an \(L^{\infty}\)-invertible function in terms of model spaces.