Composition operators for which C*φCφ and Cφ + C*φ commute

Sungeun Jung, Yoenha Kim, Eungil Ko

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We defineΘ:= {T ∈ L(H): [T*T, T +T*] = 0}. In this paper, we characterize composition operators Cφ 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 Cφ ∈ 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 Cφ with c = 0 and C*φ ∈ Θ are normal. Moreover, we show that if φ (0) = 0 and Cφ is not normal, then Cφ ∈ 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 Cφ in Θ where φ is an analytic selfmap into D. In particular, this operator has the single-valued extension property.

Original languageEnglish
Pages (from-to)1608-1625
Number of pages18
JournalComplex Variables and Elliptic Equations
Issue number12
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 Taylor & Francis.


  • Composition operators
  • Theta class


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