Spectra of compact composition operators over bounded symmetric domains (Q1772907)

Let \(H^2(d)\) denote the Hardy space of a bounded symmetric domain \(D\in{\mathbb C}^n\) in its standard Harish-Chandra realization, and let \(A_\alpha^p(D)\) be the weighted Bergman space with \(p\geq 1\) and \(\alpha<\varepsilon_D,\) where \(\varepsilon_D\) is a critical value depending on \(D\). Suppose that \(\phi:D\to D\) is holomorphic. The paper shows that if the composition operator \(C_\phi\) defined by \(C_\phi(f)=f\circ\phi\) is compact (or, more generally, power compact) on \(H^2(d)\) or \(A^p_\alpha\), then \(\phi\) has a unique fixed point \(z_0\) in \(D\). It is then proved that the spectrum of \(C_\phi\) as an operator on these function spaces is precisely the set consisting of \(0,1,\) and all possible products of eigenvalues of \(\phi(z_0)\). These results extend previous work by \textit{J.~G. Caughran} and \textit{H.~I. Schwartz} [Proc. Am. Math. Soc. 51, 127--130 (1975; Zbl 0309.47003)] and \textit{B.~D. MacCluer} [Analysis 4, 87--103 (1984; Zbl 0582.32009)]. As a corollary, MacCluer's previous spectrum results on the unit ball \(B_n\) can be extended to \(H^p(\Delta^n)\) (not only for \(p=2\), but for all \(p>1\)) and \(A_\alpha^p(\Delta^n)\) (for \(p\geq 1\)), where \(\Delta ^n\) is the polydisk in \({\mathbb C}^n\).