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

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.2.18$