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?
Asked
Active
Viewed 114 times
3
-
1This question has a detailed answer: https://math.stackexchange.com/questions/899230/difference-between-an-isometric-operator-and-a-unitary-operator-on-a-hilbert-spa – Yly Jan 19 '21 at 03:18
1 Answers
3
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 $$ S\sum_kc_ke_k=\sum_kc_ke_{k+1}. $$
Martin Argerami
- 217,281