Vol. 21, No. 2, 1967

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On operators whose Fredholm set is the complex plane

Marinus A. Kaashoek and David Clark Lay

Vol. 21 (1967), No. 2, 275–278

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 λT 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,

Mathematical Subject Classification
Primary: 47.45
Received: 28 March 1966
Published: 1 May 1967
Marinus A. Kaashoek
David Clark Lay