Let \(D\) be a bounded domain in \(\widehat{C}\) (the extended complex plane), whose boundary \(\Gamma\) is a rectifiable Jordan curve, and let \(D^*=\widehat{C}\setminus \overline{D}\). By \(B_{2}(D)\) the authors denote the Besov space of all analytic functions in \(D\) with square integrable derivative. Let \(\Delta\) denote the open unit disk. Assume that \(\psi: \Delta^*\to D^*\) is a conformal mapping such that \(\psi(\infty)=\infty\) and \(\psi^\prime(\infty)>0\). The authors introduce the Faber operator \[ Tf(z)=\frac{1}{2\pi i}\int_{\partial \Delta}\frac{f(w)\psi^\prime(w)}{\psi(w)-z}\, d w, \quad z\in D\cup D^*,\quad f\in B_{2}(\Delta), \] and prove that the following are equivalent:
(1) \(T\) is a bounded isomorphism from \(B_{2}(\Delta)\) onto \(B_{2}(D)\);
(2) \(T\) is a bounded operator from \(B_{2}(\Delta)\) into \(B_{2}(D^*)\) with \(\|T\|<1\);
(3) \(\Gamma\) is a quasi-circle.
In particular, this result shows that the conjecture, stated in [J. M. Anderson, Lect. Notes Math. 1105, 1–10 (1984; Zbl 0578.41024)], that, for every \(1<p<\infty\) and every \(D\) with a rectifiable Jordan curve as boundary, \(T\) is a bounded isomorphism from \(B_{p}(\Delta)\) onto \(B_{p}(D)\), is not true in the case \(p=2\).