Let T be a closed linear
operator with domain and range in a complex Banach space X. The Fredholm set
Φ(T) of T is the set of complex numbers λ such that λ−T is a Fredholm operator. If
the space X is of finite dimension then, obviously, the domain of T is closed and
Φ(T) is the whole complex plane C. In this paper it is shown that the converse is also
true. When T is defined on all of X this is a well-known result due to Gohberg and
Krein.
Examples of nontrivial closed operators with Φ(T) = C are the operators whose
resolvent operator is compact. A characterization of the class of closed linear
operators with a nonempty resolvent set and a Fredholm set equal to the complex
plane will be given,
