TY - JOUR

T1 - Composition operators for which C*φCφ and Cφ + C*φ commute

AU - Jung, Sungeun

AU - Kim, Yoenha

AU - Ko, Eungil

N1 - Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). The second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology) [NRF-2011-355-C00005].
Publisher Copyright:
© 2014 Taylor & Francis.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Composition operators

KW - Theta class

UR - http://www.scopus.com/inward/record.url?scp=84892977471&partnerID=8YFLogxK

U2 - 10.1080/17476933.2013.868449

DO - 10.1080/17476933.2013.868449

M3 - Article

AN - SCOPUS:84892977471

SN - 1747-6933

VL - 59

SP - 1608

EP - 1625

JO - Complex Variables and Elliptic Equations

JF - Complex Variables and Elliptic Equations

IS - 12

ER -