The spectra of Fredholm operators in locally convex spaces

Abstract

1. Notation and definitions. In this paper necessary and sufficient conditions are found for the spectrum of a Fredholm operator in a locally convex space (always taken to be Hausdorff) to lie on the non-negative real axis of the complex plane. Some results of Grothendieck(2) allow us to obtain the results in this general form; an interesting special case is the trace-class of operators in a general Banach space. We also deal with the case of Hilbert–Schmidt operators in a Hilbert space.