# Journal of Operator Theory

Volume 86, Issue 2, Fall 2021 pp. 425-438.

Beurling type invariant subspaces of composition operators

**Authors**:
Snehasish Bose (1), P. Muthukumar (2), Jaydeb Sarkar (3)

**Author institution:**(1) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

(2) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

(3) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

**Summary: **The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: characterize $\varphi$, holomorphic self maps of $\mathbb{D}$, and inner functions $\theta \in H^\infty(\mathbb{D})$ such that the Beurling type invariant subspace $\theta H^2$ is an invariant subspace for $C_{\varphi}$. We prove the following result: $C_{\varphi} (\theta H^2) \subseteq \theta H^2$
if and only if
\[
\frac{\theta \circ \varphi}{\theta} \in \mathcal{S}(\mathbb{D}).
\]
This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.

**DOI: **http://dx.doi.org/10.7900/jot.2020may15.2286

**Keywords: **composition operators, invariant subspaces, inner functions, Blaschke products, Schur functions, singular inner functions, Hardy space

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