Geodesic
On the spectrum of a linear operator
Glasgow mathematical journal, Tome 13 (1972) no. 2, pp. 98-101

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In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory. 
Dollinger, Michael B.; Oberai, Kirti K. On the spectrum of a linear operator. Glasgow mathematical journal, Tome 13 (1972) no. 2, pp. 98-101. doi: 10.1017/S0017089500001476
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