The authors characterize the sums of idempotents in terms of ranks, traces, and dimensions of null spaces, in the Banach algebra generated by the compact operators and the identity. Another characterization is given in terms of logarithmic residues of certain analytic operator functions. Topological properties (in particular, the closure) of the set of sums of idempotents are also studied.
For the entire collection see [Zbl 1005.00052].