Spectra of some composition operators and associated weighted composition operators (Q2891158)

The paper deals with composition operators, that is, operators acting on spaces of analytic functions over the unit disc \(\mathbb D\), of type \(f\mapsto f\circ\varphi\), associated to a symbol \(\varphi:\mathbb D\to\mathbb D\). This topic has been widely studied for several decades and remains a topic to which important activities are devoted. The paper under review (whose author already made a significant contribution to this area) focuses on composition operators (and weighted composition operators) on the (classical) Hardy space \(H^2\). Concerning spectra of such operators, much is known, but there are still some gaps to be filled before one can arrive at complete characterizations (depending on the nature of the symbol). The author emphasizes two questions:NEWLINENEWLINE \(\bullet\;(Q1)\) For \(\varphi\) of hyperbolic type or either of the parabolic types, do the spectrum and essential spectrum of \(C_\varphi\) always coincide?NEWLINENEWLINE \(\bullet\;(Q2)\) Denote by \(r(\varphi)\) the spectrum radius and \(r_e(\varphi)\) the essential spectrum radius of \(C_\varphi\). It is known (see \textit{C. C. Cowen} and \textit{B. D. MacCluer} [J. Funct. Anal. 125, No. 1, 223--251 (1994; Zbl 0814.47040)] and \textit{H. Kamowitz} [J. Funct. Anal. 18, 132--150 (1975; Zbl 0295.47003)]) that the spectrum and the essential spectrum consists of a disc centered at the origin of radius \(r_e(\varphi)\) (together with isolated eigenvalues), when \(\varphi\) is of dilation type and either univalent or analytic on \(\overline{\mathbb D}\). In this situation, must every point of this disc be in the essential spectrum?NEWLINENEWLINE The author concentrates his attention on composition operators whose symbol \(\varphi\) is ``essentially linear fractional'', which means thatNEWLINENEWLINE \(\bullet\;\varphi(\mathbb D)\) is contained in a proper subdisc of \(\mathbb D\), internally tangent to \(\partial\mathbb D\) at some point \(\eta\in\partial\mathbb D\);NEWLINENEWLINE\(\bullet\;\varphi^{(-1)}(\{\eta\})\) (which is actually the set of all \(\gamma\in\mathbb D\) such that \(\eta\) belongs to the cluster set of \(\varphi\) at \(\gamma\)) consists of one element, say \(\xi\in\partial\mathbb D\);NEWLINENEWLINE\(\bullet\;\varphi'''\) extends continuously to \(\mathbb D\cup\{\xi\}\) (see also \textit{P. S. Bourdon} et al.\ [J. Math. Anal. Appl. 280, No. 1, 30--53 (2003; Zbl 1024.47008)] for more).NEWLINENEWLINE In the paper under review, the author characterizes the spectrum and essential spectrum of \(C_\varphi\) when \(\varphi\) is essentially linear fractional and answers question \((Q1)\) in the positive. As for question \((Q2)\), he shows that the characterization with the disc centered at the origin of radius \(r_e(\varphi)\) (together with isolated eigenvalues) is still valid when \(\varphi\) is essentially linear fractional with dilation type (but without additive assumption). Moreover, the answer to question \((Q2)\) is ``yes'' in this framework.NEWLINENEWLINE Let us mention some of the specific results obtained in the paper. For instance, the author obtains (Theorem 3.3) that the spectrum and the essential spectrum of \(C_\varphi\) is the spiral \(\left\{e^{-at}\mid t\geq 0\right\}\cup\{0\}\) when \(\varphi\) is essentially linear fractional (with some additive assumptions). On the other hand, the author gives characterizations of the (essential) spectrum of certain weighted composition operators with bounded weight and essentially linear fractional symbol (Theorems 4.4, 4.5, 4.6 and 4.7).NEWLINENEWLINE It is worth mentioning that the results are illustrated several times by some (well detailed) ``concrete'' examples. The author finishes the paper with a series of open questions.