Local spectral theory of endomorphisms of the disk algebra

Let A() denote the disk algebra. Every endomorphism of A() is induced by some ϕ ∈ A() with ‖ϕ‖ ≤ 1. In this paper, it is shown that if ϕ is not an automorphism of and ϕ has a fixed point in the open unit disk then the endomorphism induced by ϕ is decomposable if and only if the fixed set of ϕ is singleton. Further, we determine the local spectra of the endomorphism induced by ϕ in the cases when the fixed set of ϕ either includes unit circle or is a singleton