Spectra of composition operators with symbols in \(\mathcal S(2)\) (Q2835224)

The paper deals with composition operators, i.e., operators acting on spaces of analytic functions over the unit disc \(\mathbb D\) of the form \(f\mapsto f\circ\varphi\), associated to a symbol \(\varphi:\mathbb D\to\mathbb D\). This topic has been widely studied for several decades, but there are still many open questions around the spectra of such operators even in the (classical) framework of the Hardy space \(H^2\). The paper under review focuses on composition operators associated to symbols \(\varphi\) belonging to the so-called class \({\mathcal S}^2\) introduced by \textit{T. Kriete} and \textit{J. Moorhouse} [Trans. Am. Math. Soc. 359, No. 6, 2915--2944 (2007; Zbl 1115.47023)]: roughly speaking, the ``sufficient-data'' class \({\mathcal S}^2\) consists of symbols \(\varphi\) having limited contact with \(\partial{\mathbb D}\) that is both of order \(2\) and \({\mathcal C}^2\) (the rigorous definition is stated with the help of Aleksandrov-Clark measures). This paper is a sequel of the paper [J. Oper. Theory 67, No. 2, 537--560 (2012; Zbl 1262.47036)] of the same author.NEWLINENEWLINEThis interesting paper begins with a valuable exposition of results on spectra and essential spectra of composition operators, and contains various results. Let us emphasize Theorem 4.4. which describes the spectrum and essential spectrum of \(C_\varphi\) where \(\varphi\in{\mathcal S}(2)\), according to the nature of the Denjoy-Wolff point. Consider the symbol NEWLINE\[NEWLINE\varphi(z)=\frac{2z^2-z-2}{2z^2-3}.NEWLINE\]NEWLINE Theorem 4.4. shows that both \(\sigma(C_\varphi)\) and \(\sigma_e(C_\varphi)\) are equal to \(\overline{D}(0,1/3)\cup[0,1]\) (lollipop-shaped), whereas \(\varphi\) is of parabolic non-automorphism type. Hence the author obtains a negative answer to a conjecture of \textit{C. C. Cowen} [J. Oper. Theory 9, 77--106 (1983; Zbl 0504.47032)] stating that spectrum of such symbol could lie between two spirals. Some other applications and examples are given at the end of the paper. At last: two open questions are stated (naturally related to Theorem 4.4): For \(\varphi\) of hyperbolic type or of parabolic type, do the spectrum and essential spectrum of \(C_\varphi\) always coincide? Let \(\varphi\) be an non-automorphic analytic selfmap of \({\mathbb D}\) having its Denjoy-Wolff point in \({\mathbb D}\). Does the essential spectrum consist of a disk (possibly degenerate) of radius less than \(1\)?