Spectral theory for algebraic combinations of Toeplitz and composition operators (Q732042)

Let \(\mathbb{D}\) denote the open unit disk in the complex plane and \(H^2\) the Hardy space of all holomorphic functions on \(\mathbb{D}\) with square summable power series coefficients. An analytic self-map \(\varphi\) of \(\mathbb{D}\) induces through composition a linear composition operator \[ C_{\varphi}: H^2 \to H^2, \; f \mapsto f \circ \varphi. \] It is well known that such an operator is bounded. \(H^2\) can be considered as the subspace of \(L^2\) spanned by \(\{e^{in\theta}:n\geq 0\}\) if any \(H^2\)-function is identified with its (almost everywhere existing) non-tangential limit function on the circle. Thus, any bounded, measurable, complex-valued function \(w\) on \(\partial \mathbb{D}\) gives a Toeplitz operator \[ T_w: H^2 \to H^2, \; f \mapsto P(wf), \] where \(P\) is the orthogonal projection of \(L^2\) onto \(H^2\). Motivated by the article [Trans.\ Am.\ Math.\ Soc.\ 359, No.\,6, 2915--2944 (2007; Zbl 1115.47023)] by \textit{T.\,Kriete} and \textit{J.\,Moorhouse}, the authors determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The approach to do this is to realize the \(C^*\)-algebra that they generate as an extension of the compact operators by a concrete \(C^*\)-algebra whose invertible elements are easily characterized.