Ben Hur Eidt | Searcharxiv

arXiv2024

Beurling and Model subspaces invariant under a universal operator

In this article, we characterize the Beurling and Model subspaces of the Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator $C_{ϕ_a}f=f\circϕ_a$, where $ϕ_a(z) = az + 1 - a$ for $a \in (0,1)$ is an affine self-map of the open unit disk $\mathbb{D}$. These operators have universal translates (in the sense of Rota) and have attracted attention recently due to their connection with the Invariant Subspace Problem (ISP) and the classical Cesàro operator.

math.FA↗

arXiv2023

Minimal invariant subspaces for an affine composition operator

The composition operator $C_{ϕ_a}f=f\circϕ_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for $C_{ϕ_a}$ is one-dimensional. These minimal invariant subspaces are always singly-generated $ K_f := \overline{\mathrm{span} \{f, C_{ϕ_a}f, C^2_{ϕ_a}f, \ldots \}}$ for some $f\in H^2(\mathbb{D})$. In this article we characterize the minimal $K_f$ when $f$ has a nonzero limit at the point $1$ or if its derivative $f'$ is bounded near $1$. We also consider the role of the zero set of $f$ in determining $K_f$. Finally we prove a result linking universality in the sense of Rota with cyclicity.

math.FA↗