Spectra of composition operators on the Bloch and Bergman spaces (Q701372)

Let \(U\) be the unit disc in the complex plane and \(\varphi\) be an analytic map from \(U\) to itself. In [J. Am. Math. Soc. 10, 299-325 (1997; Zbl 0870.30018)], \textit{P. S. Bourdon} and \textit{J. H. Shapiro} give the essential spectral radius formula on the Hardy space \(H^p\), for any analytic \(\varphi\) and \(0<p<\infty\). Using the Calderón theory of complex interpolation, the authors obtain an analogous result on the Bergman space \(A^p\), namely that for \(1<p<\infty\), \[ \bigl( r_{e,A^p}(C_\varphi) \bigr)^p = \bigl( r_{e,A^2}(C_\varphi) \bigr)^2. \] They also obtain the spectrum of the composition operator \(C_\varphi\) on the Bergman space, Bloch space, BMOA and Hardy space for \(\varphi\) univalent, not an automorphism, with fixed point in \(U\).