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Consider two unitary operators $A$ and $B$ acting on a Hilbert space.

Let's assume that $\forall |\psi \rangle$ (vector living in this Hilbert space), we have: $\langle \psi | A | \psi \rangle = \langle \psi | B | \psi \rangle$.

I am almost sure that it implies $A=B$, but I do not remember how to show it simply. Am I correct? How can it be easily derived?

StarBucK
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    The answer depends on whether the underlying field is $\Bbb{R}$ or $\Bbb{C}$. Do you have such a restriction? – Nick F Dec 16 '22 at 17:17
  • Use polarization. – anomaly Dec 16 '22 at 17:23
  • This post should not have been closed, because of the following remaining question. On a real Hilbert space, $\langle \psi | A | \psi \rangle = \langle \psi | B | \psi \rangle$ for all $ | \psi \rangle$ does not imply $A=B$ in general. But does it imply $A=B$ under the additional assumption that $A$ and $B$ are unital? (I doubt it but have no counterexample). @anomaly – Anne Bauval Dec 16 '22 at 22:24

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