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