Structure of Generalized bi-circular idempotents and isometric reflections on $C^1[0,1]$

Abstract

Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \|f\|_σ = |f(0)| +\|f'\|_\infty$. Let $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $λ_1P_1 + λ_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $λ_1, λ_2$, if and only if either $λ_1 + λ_2 = 0$, or $λ_1P_1 + λ_2P_2$ is an isometry for all such complex numbers $λ_1, λ_2$. In the former case, the collection $\{P_1, P_2\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.