TY - JOUR
T1 - Commutators of weighted composition operators
AU - Jung, Sungeun
AU - Kim, Yoenha
AU - Ko, Eungil
N1 - Funding Information:
The authors wish to thank the referee for a careful reading and valuable comments for the original draft. 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).
PY - 2014/6
Y1 - 2014/6
N2 - In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of such that φ(ζ) and ψ(ζ) belong to ∂ for some ζ ∈ ∂ and u, v ∈ H∞ are continuous on ∂ with u(ζ)v(ζ) ≠ 0, then [{W*v, psi, Wu,φ] is compact on H2 if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is 1=w where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H2. We verify that if φ is an analytic self-map of with Denjoy-Wolff point b ∈ and u ∈ H∞\{0}, then every weighted composition operator in the commutant {Wu, φ}′ has {f ∈ H2 : f(b) = 0} as its nontrivial invariant subspace.
AB - In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of such that φ(ζ) and ψ(ζ) belong to ∂ for some ζ ∈ ∂ and u, v ∈ H∞ are continuous on ∂ with u(ζ)v(ζ) ≠ 0, then [{W*v, psi, Wu,φ] is compact on H2 if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is 1=w where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H2. We verify that if φ is an analytic self-map of with Denjoy-Wolff point b ∈ and u ∈ H∞\{0}, then every weighted composition operator in the commutant {Wu, φ}′ has {f ∈ H2 : f(b) = 0} as its nontrivial invariant subspace.
KW - Commutant
KW - Commutator
KW - Compact operator
KW - Weighted composition operator
UR - http://www.scopus.com/inward/record.url?scp=84903528826&partnerID=8YFLogxK
U2 - 10.1142/S0129167X14500530
DO - 10.1142/S0129167X14500530
M3 - Article
AN - SCOPUS:84903528826
SN - 0129-167X
VL - 25
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 6
M1 - 1450053
ER -