How Small Can a Sum of Idempotents Be

The issue discussed in this paper is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents $$p_1,\ldots ,p_n$$ in a complex algebra or Banach algebra, is it possible that their sum $$p_1+\cdots +p_n$$ is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem comes from earlier work by the authors on the generalization of the logarithmic residue theorem from complex function theory to higher (possibly infinite) dimensions.