Characterization of Unitary Operators
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Let $H$ be a Hilbert space.
Let $A \in \map B H$ be a bounded linear operator.
Then the following are equivalent:
- $(1): \quad A$ is a unitary operator
- $(2): \quad A^* A = A A^* = I$, where $A^*$ denotes the adjoint of $A$, and $I$ denotes the identity operator
- $(3): \quad A$ is a normal isometry