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 $$p_1,\ldots ,p_n$$ p 1 , … , p n in a complex algebra or Banach algebra, is it possible that their sum $$p_1+\cdots +p_n$$ p 1 + ⋯ + 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.

中文翻译：

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幂等项的总和可以有多小？

本文讨论的问题是：幂等项的总和可以有多小？这里的小是根据幂零或准幂等来理解的。因此问题是：给定复数代数或巴拿赫代数中的幂等项 $$p_1,\ldots ,p_n$$ p 1 , ... , pn ，它们的和是否可能为 $$p_1+\cdots +p_n$$ p 1 + ⋯ + pn 是拟幂零还是（偶数）幂零（按一定顺序）？考虑这个问题的动机来自作者早期的工作，他们将对数残差定理从复函数理论推广到更高（可能是无限）维度。