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 -