Generalized idempotents on the space of analytic functions with bounded derivatives

Let $X$ be a complex normed space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$. A collection $\mathcal{C} = \{P_1, P_2\}$ of nonzero distinct orthogonal ($P_1P_2 = P_2P_1 = 0$) 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 $\lambda_1P_1 + \lambda_2P_2$ is a surjective isometry on $X$. This generalizes the notion of generalized bi-circular projections on Banach spaces introduced by Fo\v{s}ner, Ili\v{s}evi\'{c} 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.