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Compact composition operators on BMOA and the Bloch space. (English) Zbl 1194.47038

Let Δ denote the unit disc in the complex plane and let H(Δ) denote the set of all functions holomorphic on Δ. For a point aΔ, let

σ a (z)=a-z 1-a ¯z·

The spaces BMOA and the Bloch space are defined by

BMOA=fH(Δ):f * 2 =sup aΔ lim r1 1 2π 0 2π |f(σ a (re iθ ))-f(a)| 2 dθ<

and

=f H(Δ):f ** = sup zΔ {|f ' (z)|(1-|z| 2 )} < ·

Then f BMOA =f * +|f(0)| and f =f ** +|f(0)|. Let ϕH(Δ) such that ϕ(Δ)Δ, and, for fH(Δ), let C ϕ (f)=fϕ. Let X denote either of the spaces BMOA or . The authors prove that the composition operator C ϕ is a compact operator on the space X if and only if lim n ϕ n X =0. For the space BMOA, this improves a result of the first author [Sci. China, Ser. A 50, No.  7, 997–1004 (2007; Zbl 1126.30023)].


MSC:
47B38Operators on function spaces (general)
30H30Bloch spaces
30H35BMO-spaces
30J99Function theory on the disc