Isomorphisms of the disc algebra and inverse Faber sets
Let K denote the closed interior of the Jordan curve \(\Gamma\), and A(K) the Banach algebra of functions f(z), analytic in \(K^ 0\) (the interior of \(\Gamma)\) and continuous on K, with the supremum norm. In the case when \(\Gamma\) is the unit circle, we write \(K=\Delta\) and \(A(\Delta)=A\). Let \(z=\psi (w)\) be the unique conformal mapping of \({\mathbb{C}}\setminus \Delta\) onto \({\mathbb{C}}\setminus K\) with \(\psi (\infty)=\infty\), \(\psi '(\infty)=\alpha >0\); \(\psi\) has a continuous extension to \(\overline{{\mathbb{C}}\setminus \Delta}\). Assume that \(\alpha =1\). The Faber polynomials associated with K are defined by \[ \psi '(w)/(\psi (w)- z)=\sum^{\infty}_{n=0}F_ n(z)\omega^{-(n+1)}. \] The Faber operator T is the unique linear automorphism of the space of all polynomials, defined by: \(T: w^ n\to F_ n(z)\). Its inverse \(T^{-1}\) is the inverse Faber operator. If T is a bounded operator, we say that K is a Faber set. In this case, T has an extension, as a bounded operator, to the disc algebra A. If both T and \(T^{-1}\) are bounded operators, the authors call K an inverse Faber set. It was proved by the reviewer in 1972 that if \(\Gamma\) is of bounded rotation, then K is a Faber set. In the present paper the authors prove that if, in addition, \(\Gamma\) is free of cusps, then K is an inverse Faber set. This is a surprising result of very great interest. The authors also construct a set K of bounded boundary rotation (with a cusp) which is not an inverse Faber set. One suspects that the positive result remains valid even if the boundary of K is not a Jordan curve, as long as \(\psi\) (w) is continuous for \(| w| \geq 1\), and \(\psi (e^{it})\) is a path of bounded rotation, without cusps.
scientific article; zbMATH DE number 3935387