A note on Faber operator

For a rectifiable Jordan curve Γ with complementary domains D and D*, Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces B _p (1 < p < ∞) of analytic functions in the unit disk and in the inner domain D, respectively. We point out that the conjecture is not true in the special case p = 2, and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.