Abstract: 
\def\A{\cal A}
Let $\A$ be a Banach algebra without nonzero finite dimensional ideals. 
Then every compact semiderivation on $\A$ is a quasinilpotent operator
mapping $\A$ into its radical.

1. Department of Mathematics,
   University of Maribor
   PEF, Koroška 160
   2000 Maribor, Slovenia
   
2. Institute of Mathematics and
   Mechanics of National Academy of
   Sciences of Azerbaijan
   F. Agaev St. 9, Baku AZ1141
   Azerbaijan