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

Compact composition operators on ${H}^{\infty}\left(G\right)$, 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 ${H}^{\infty}\left(D\right)$, where D is the unit disc, and determine their spectra.