Characterization of Unitary Operators

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Theorem

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


Proof


Sources