Generalized idempotents on the space of analytic functions with bounded derivatives

Let $X$ be a complex normed space. A map $P:X\to X$ is called idempotent if $P^2=P$. A collection $\mathcal{C}=\{P_1,P_2\}$ of nonzero distinct orthogonal ($P_1{P}_2=P_2{P}_1=0$) idempotent maps on $X$ is said to be a family of generalized bi-circular idempotents if there exist distinct unit modulus complex numbers ${\lambda }_1,{\lambda }_2$ such that $P_1+P_2=I$ (identity operator on $X$) and ${\lambda }_1{P}_1+{\lambda }_2{P}_2$ is a surjective isometry on $X$. This generalizes the notion of generalized bi-circular projections on Banach spaces introduced by Fošner, Ilišević and Li \cite{MDC} to nonlinear maps. In this paper, we describe the structure of generalized bi-circular idempotents over the space of analytic functions on the open unit disk with bounded derivatives.