Bounded Linear Transformation is Isometry iff Adjoint is Left-Inverse

Theorem

Let $\HH, \KK$ be Hilbert spaces.

Let $A \in \map B{\HH, \KK}$ be a bounded linear transformation.

Then $A$ is an isometry if and only if:

$A^*A = I_\HH$

where $A^*$ denotes the adjoint of $A$, and $I_\HH$ the identity operator on $\HH$.

Proof

Let $g, h \in \HH$. Then by the definition of adjoint:

${\innerprod {A g} {A h} }_\KK = {\innerprod {A^* A g} h}_\HH$

From the uniqueness of the adjoint, it follows that:

${\innerprod {A g} {A h} }_\KK = {\innerprod g h}_\HH$

holds if and only if $A^*A = I_\HH$.

Hence the result by definition of isometry.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.17$