Vol. 21, No. 2, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
On operators whose Fredholm set is the complex plane

Marinus A. Kaashoek and David Clark Lay

Vol. 21 (1967), No. 2, 275–278
Abstract

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
Milestones
Received: 28 March 1966
Published: 1 May 1967
Authors
Marinus A. Kaashoek
David Clark Lay