Automorphisms and generalized projections on spaces of analytic functions

We present complete classifications of automorphisms of two closed subalgebras of the bounded analytic functions on the open unit disc $\mathbb{D}$, namely, the subalgebra of functions vanishing at the origin, and the subalgebra of functions whose first derivative vanishes at the origin. The later subalgebra is known as the Neil algebra. We also characterize generalized tri-circular projections on $H^{p}(\mathbb{D})$ and $H^{p}(\mathbb{D}^2)$, $1\leq p \leq \infty$, $p\neq 2$.