Is it possible that an isometric (bounded) operator ${\rm T}: \mathcal H_1 \to \mathcal H_2$ with norm $\|\rm T\| = 1 $ is not unitary, i.e. $\rm T^*T = I \neq TT^*$?
If yes, could you please provide some examples?
2026-03-30 09:53:48.1774864428
Isometric linear operators on Hilbert spaces
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Yes. The canonical example is the unilateral shift.
Take any separable Hilbert space $H$, and choose some orthonormal basis $\{e_n\}_{n\in\mathbb N}$ for it. Then define $$ S\sum_kc_ke_k=\sum_kc_ke_{k+1}. $$