Vol. 64, No. 1, 1976

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Spectral approximation theorems in locally convex spaces

Volker Wrobel

Vol. 64 (1976), No. 1, 273–281

We present some results on collectively compact operator approximation theory in locally convex Hausdorff spaces (l.c.s.). The notion of a collectively compact family of operators acting on a Banach space has been introduced by Anselone and Palmer in connection with the numerical solution of integral equations. Meanwhile collectively compact families of operators have been studied in general topological vector spaces. In contrast to those investigations dedicated to the characterization of collectively compact families of operators the present paper focuses on spectral approximation theorems in l.c.s. similar to those given by Anselone and Palmer in the case of Banach spaces. In doing this it turns out that the notion of the spectrum, which causes no problems in Banach algebra theory, entails some difficulty. A way out is indicated by using notions and tools of locally convex algebra theory.

Mathematical Subject Classification 2000
Primary: 47A55
Secondary: 47A60
Received: 28 January 1975
Revised: 12 January 1976
Published: 1 May 1976
Volker Wrobel