Abstract
The present paper is concerned with 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. These sums are characterized in terms of ranks, traces and dimensions of null spaces. Another, quite different, characterization is given in terms of logarithmic residues, i.e., contour integrals of logarithmic derivatives, of certain analytic operator functions. The functions in question have values in the (non-closed) subalgebra generated by the identity and the finite rank operators. Topological properties of the set of sums of idempotents are considered too.
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Bart, H., Ehrhardt, T., Silbermann, B. (2002). Sums of Idempotents in the Banach Algebra Generated by the Compact Operators and the Identity. In: Böttcher, A., Gohberg, I., Junghanns, P. (eds) Toeplitz Matrices and Singular Integral Equations. Operator Theory: Advances and Applications, vol 135. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8199-9_4
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DOI: https://doi.org/10.1007/978-3-0348-8199-9_4
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