Local spectral theory of endomorphisms of the disk algebra

Abstract

LetApDqdenote the disk algebra. Every endomorphism ofApDqis inducedby someφPApDqwith}φ}≤1. In this paper, it is shown that ifφis not an automorphismofDandφhas a fixed point in the open unit disk then the endomorphism induced byφis decomposable if and only if the fixed set ofφis singleton. Further, we determine thelocal spectra of the endomorphism induced byφin the cases when the fixed set ofφeitherincludes unit circle or is a singleton.