Fredholmness of a linear combination of operators

Although it is well known that Fredholmness of the linear combination αP+βQ, α,β∈C∖{0}, α+β≠0, does not depend on the choice of scalars if P,Q∈B(H) are idempotents, no necessary and sufficient conditions for Fredholmness of this linear combination are yet known, except for the special case when P and Q are orthogonal projectors. In this paper, using a completely different approach and some results on completion problems of operator matrices, we give necessary and sufficient conditions for Fredholmness of a sum of two idempotents. Also, we will discuss the more general question when the sum of two operators is a Fredholm operator and consider some special cases when Fredholmness of a linear combination of two operators is independent of the choice of the scalars. On the other hand some special classes of operators for which a linear combination of two operators depends of the choice of the scalars are listed. A new proof of a well-known result is given.