| 1. | Thus all sets of Kraus operators are related by partial isometries.
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| 2. | The concept of partial isometry can be defined in other equivalent ways.
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| 3. | A partially defined isometric operator with closed domain is called a partial isometry.
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| 4. | Fix two such partial isometries for the argument.
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| 5. | Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.
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| 6. | A partial isometry " V " has a unitary extension if and only if the deficiency indices are identical.
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| 7. | "V " can be extended to a partial isometry acting on all of \ mathcal { H }.
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| 8. | In general, a partial isometry may not be extendable to a unitary operator and therefore a quasinormal operator need not be normal.
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| 9. | The operator " U " must be weakened to a partial isometry, rather than unitary, because of the following issues.
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| 10. | Passing to the non-commutative setting, this motivates " Krein's formula " which parametrizes the extensions of partial isometries.
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