Finding the Adjoint of an Infinite Hilbert Space Operator

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SUMMARY

The discussion focuses on finding the adjoint of the operator O defined on an infinite Hilbert space, where Oei = ei+1 for positive integers i. The adjoint operator O* is determined to be O*ei = ei-1. The participants confirm that the composition of the operators OO* results in the identity operator, as demonstrated by OO*ei = ei, thus establishing that OO* = I. Additionally, the same approach can be applied to show that O*O = I.

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Homework Statement


Hi guys

I have the following operator on an infinite Hilbert space: Oei=ei+1, where i is a positive integer and ei 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*ei=ei-1.

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

Thanks for any input.
 
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You've done the hard work, that's trivial now! For any basis vector ei, O*ei= ei_1 so OO*ei= Oei-1= e(i-1)+1= ei.

You can show that O*O = 1 in exactly the same way.
 

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