This is the first of two survey articles based on lectures given by the two authors as part of the Special Year in Operator Theory held at Indiana University during the 1985-86 academic year. This first volume is primarily concerned with the study of single operators, while the second will focus on algebras of bounded operators on Hilbert space. The idea behind these lectures is to develop a topic of current interest in operator theory, departing from a point accessible to advanced graduate students and proceeding to its latest stage. The authors of these surveys are acknowledged experts who have taken on the role of advocate of their discipline.
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