## Abstract

We defineΘ:= {T ∈ L(H): [T*T, T +T*] = 0}. In this paper, we characterize composition operators C_{φ} and their adjoints C*_{φ} which belong to Θ, where the maps (Formula presented) are linear fractional selfmaps of the open unit disk D into itself. If φ is an automorphism of D or c = 0, then the case for C_{φ} ∈ Θis precisely when it is normal. When c = 0, we also prove that if C*_{φ} ∈ Θ, then either b = 0 or (Formula presented), which implies that the only binormal composition operators C_{φ} with c = 0 and C*_{φ} ∈ Θ are normal. Moreover, we show that if φ (0) = 0 and C_{φ} is not normal, then C_{φ} ∈ Θ implies that (Formula presented) and d/a is neither real nor purely imaginary, while C*_{φ} ∈ Θ ensures that (Formula presented) and d/a is real. Finally, we study composition operators C_{φ} in Θ where φ is an analytic selfmap into D. In particular, this operator has the single-valued extension property.

Original language | English |
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Pages (from-to) | 1608-1625 |

Number of pages | 18 |

Journal | Complex Variables and Elliptic Equations |

Volume | 59 |

Issue number | 12 |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© 2014 Taylor & Francis.

## Keywords

- Composition operators
- Theta class

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