# Characterization of Unitary Operators

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## Theorem

Let $\HH$ be a Hilbert space.

Let $A$ be a bounded linear operator on $\HH$.

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

## Proof

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## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $\text {II}.2.18$