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:<math>N N^* = \begin{bmatrix} UP^2U^* + D_{U^*} P^2 D_{U^*} & -D_{U^*}P^2 U \\ -U^* P^2 D_{U^*} & U^* P^2 U \end{bmatrix}.</math>
Because ''UP = PU'' and ''P'' is self adjoint, we have ''U*P = PU*'' and ''D<sub>U*</sub>P = D<sub>U*</sub>P''. Comparing entries then shows ''N'' is normal. This proves quasinormality implies normality.
For a counter example that shows the converse is not true, consider again the unilateral shift ''A''. The operator ''B'' = ''A'' + ''s'' for some scalar ''s'' remains subnormal. But if ''B'' is quasinormal, a straightforward calculation shows that ''A*A = AA*'', which is a contradiction.
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