Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure.