A note on the nullspace of linear combinations of scalar idempotent operators on Banach space

摘要

By the properties of scalar idempotent operators and their linear combinations on Banach space, we get the isomorphism theory of the corresponding nullspace independently of the choice of combination coefficients. As an application, it not only derive the conclusion of J. J. Koliha and V. Rako cevi'c in 2007 about Fredholmness of linear combinations of two idempodent operators on Banach space, which is independent of the choice of combination coefficients, but also extend the discussion to any Banach space.