Most of the progress which has been made in the study of bounded linear operators on Hilbert space is restricted to operators with thin spectrum or operators which can be written as a direct sum of such operators.Our aim in this paper is to begin a systematic study of a class of operators at the opposite extreme, namely those which possess an open set of eigenvalues.We relate the study of such operators to certain problems in complex
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Cowen et al. (1978) studied this question.