Abstract
Let T T be a bounded linear operator on a Hilbert space and assume T T has thin spectrum. When is T T similar to a normal operator? This problem is studied in a variety of situations and sufficient conditions are given in terms of characteristic functions, resolvents, spectral sets, and spectral resolutions. By contrast, the question “When is T T normal?” has a relatively simple answer since in that case a necessary and sufficient condition can be given in terms of the resolvent alone.
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