| 1. | An example of such an operator is a normal operator.
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| 2. | The residual spectrum of a normal operator is empty.
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| 3. | There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces.
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| 4. | A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.
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| 5. | Normal operators are characterized by the spectral theorem.
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| 6. | The operator norm of a normal operator equals its numerical radius and spectral radius.
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| 7. | This is the spectral theorem for normal operators.
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| 8. | Every normal operator is subnormal by definition, but the converse is not true in general.
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| 9. | Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces.
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| 10. | It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator.
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