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Hi guys

I have the following operator on an infinite Hilbert space: Oe_{i}=e_{i+1}, whereiis a positive integer and e_{i}are the orthonormal vectors that span the space. I have to find the adjoint of this operator.

For two arbitrary vectors x and z, I have found (Ox,z), and I have to find the O* (the adjoint) such that (Ox,z)=(x,O*z). This gives me that O*e_{i}=e_{i-1}.

Now I have to show that OO* = 1. How can I go about doing that?

Thanks for any input.

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# Homework Help: Adjoint of an operator

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