Path connected components in the space of weighted composition operators on the disk algebra (Q2861055)

Let \(X\) be a Banach space of analytic functions on the unit disc \(\mathbb{D}\) of the complex plane \(\mathbb{C}\) and let \(\mathcal{C}(X)\) the space of composition operators \(C_\varphi: f \rightarrow f \circ \varphi\) that are bounded on \(X\) endowed with the operator norm induced by \(\mathcal{L}(X)\). The connected component problem for \(\mathcal{C}(X)\) consists in determining the connected component in \(\mathcal{C}(X)\) containing a given composition operator \(C_\varphi\). The case where \(X=H^2\) is the Hardy space was studied by Berkson, MacCluer, Shapiro, Sundberg, Bourdon, Gallardo-Gutierrez, González, Nieminnen, Saksman, Moorhouse and Toews. On the other hand, the case of \(X=H^\infty\) of bounded holomorphic functions on the disc was investigated by MacCluer, Ohno, Zhao, Hosokawa, Zheng and Izuchi. The path connected components of the space \(\mathcal{C}_w(H^\infty)\) of nonzero weighted composition operators \(M_u C_\varphi\) on \(H^\infty\) were determined by \textit{T. Hosokawa}, \textit{K. Izuchi} and \textit{S. Ohno} in [Integral Equations Oper. Theory 53, No.~4, 509--526 (2005; Zbl 1098.47025)], and those of the space \(\mathcal{C}_{w,0}(H^\infty)\) of noncompact weighted composition operators on \(H^\infty\) have been investigated recently by the present authors in [Trans. Am. Math. Soc. 365, No.~7, 3593--3612 (2013; Zbl 1282.47048)].NEWLINENEWLINEIn the interesting paper under review, the authors consider the structure of the space \(\mathcal{C}_{w,0}(A)\) of noncompact weighted composition operators on the disc algebra \(A=A(\overline{\mathbb{D}})\). Among other results, they prove that, if \(\varphi\) is a self map on \(\mathbb{D}\) such that \(\varphi \in A(\overline{\mathbb{D}})\), \(\|\varphi\|_{\infty}=1\), and the measure of \(\{z \in \mathbb{C} \mid |z|=1,\;|\varphi(z)|=1 \}\) is positive, then the set \(\Omega_\varphi:=\{ M_u C_\varphi \in \mathcal{C}_{w,0}(A) \mid u \in A(\overline{\mathbb{D}}) \}\) is open and closed and a path connected component of \(\mathcal{C}_{w,0}(A)\). It is also shown that the set of compact weighted composition operators on \(A(\overline{\mathbb{D}})\) is a path connected component in \(\mathcal{C}_w(A)\). Conditions to ensure that \(\Omega_\varphi\) is open and closed and a path connected component of \(\mathcal{C}_{w,0}(A)\) are given. Several examples show that the structure of the path connected components in \(\mathcal{C}_{w,0}(A)\) and \(\mathcal{C}_{w,0}(H^\infty)\) are different.