Characterization of Unitary Operators

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