Compact endomorphisms of ${\displaystyle H^{\infty }\left(D\right)}$

Compact composition operators on ${\displaystyle H^{\infty }\left(G\right)}$, where G is a region in the complex plane, and the spectra of these operators were described by D. Swanton ( Compact composition operators on B(D), Proc. Amer. Math. Soc. 56 (1976), 152-156). In this short note we characterize all compact endomorphisms, not necessarily those induced by composition operators, on ${\displaystyle H^{\infty }\left(D\right)}$, where D is the unit disc, and determine their spectra.