Path connected components in weighted composition operators on \(h^\infty\) and \(H^\infty\) with the operator norm (Q2846976)

Let \(h^\infty\) and \(H^\infty\) denote the spaces of bounded harmonic and analytic functions on the unit disk \(\mathbb{D}\), respectively. If \(u\in H^\infty\) and \(\varphi: \mathbb{D}\to\mathbb{D}\) is an analytic mapping, then \(M_u C_\varphi f = u f\circ \varphi\) defines a weighted composition operator on \(H^\infty\). For \(u\in L^\infty (\partial\mathbb{D})\), the weighted composition operator \(M_u C_\varphi\) on \(h^\infty\) is defined as NEWLINE\[NEWLINE (M_u C_\varphi f)(z) = \int_{\partial\mathbb{D}} u(\zeta)(f\circ\varphi)^*(\zeta) P_z(\zeta)\, dm(\zeta), \quad z\in\mathbb{D}, NEWLINE\]NEWLINE where \(P_z\) is the Poisson kernel for the point \(z\) and \(m\) is the normalized Lebesgue measure on \(\partial\mathbb{D}\).NEWLINENEWLINEThe authors determine the (operator norm) path connected components in the sets of noncompact weighted composition operators on \(h^\infty\) and \(H^\infty\). In particular, they answer questions formulated by \textit{T. Hosokawa} and the first and third authors [Integral Equations Oper. Theory 53, No. 4, 509--526 (2005; Zbl 1098.47025)].