Characterization of Unitary Operators

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Then the following are equivalent:


 * $(1): \qquad A$ is a unitary operator
 * $(2): \qquad A^*A = AA^* = I$, where $A^*$ denotes the adjoint of $A$, and $I$ denotes the identity operator
 * $(3): \qquad A$ is a normal isometry