Isometry iff Adjoint is Left-Inverse

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Theorem

Let $H, K$ be Hilbert spaces.

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


Then $A$ is an isometry iff:

$A^*A = I_H$

where $A^*$ denotes the adjoint of $A$, and $I_H$ the identity operator on $H$.


Proof

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

$\left\langle{Ag, Ah}\right\rangle_K = \left\langle{A^*Ag, h}\right\rangle_H$


From the uniqueness of the adjoint, it follows that:

$\left\langle{Ag, Ah}\right\rangle_K = \left\langle{g, h}\right\rangle_H$

holds iff $A^*A = I_H$.


Hence the result by definition of isometry.

$\blacksquare$


Sources