V. A. Anjali | Searcharxiv

arXiv2026

Singular inner eigenfunctions of composition operators

This paper characterizes all the singular inner eigenfunctions of the composition operators $C_ϕ$ that arise from discrete measures, when $ϕ$ is an automorphism of unit disk. By establishing a connection between Beurling and model invariant subspaces, we classify all the inner functions so that the corresponding Beurling subspace is invariant under the composition operators induced by non-elliptic automorphisms. This classification involves solving the eigenfunction equation for the composition operator. Further, we present some applications of the above-mentioned connection.

math.FA↗

arXiv2024

Composition operators between Beurling subspaces of Hardy space

V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators $C_ϕ$ on the Hardy space $H^2$ and nonzero singular measures $μ_1$, $μ_2$ on the unit circle such that $C_ϕ({S_{μ_1}} H^2)\subseteq {S_{μ_2}} H^2,$ where $S_{μ_i}$ denotes the singular inner function corresponding to the measure $μ_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $ϕ$ of the unit disk $\mathbb{D}$ and inner functions $θ_1, θ_2$ such that $C_ϕ(θ_1 H^p)\subseteq θ_2 H^p,$ for $p>0$. Emphasis is given to Blaschke products and singular inner functions as a special case. We also give an another measure-theoretic characterization to above question when $ϕ$ is an elliptic automorphism. For a given Blaschke product $θ$, we discuss about finding all self maps $ϕ$ such that $θH^p$ is invariant under $C_ϕ$.

math.FA↗