# Homework Help: Adjoint of an operator

1. Jan 10, 2010

### Niles

1. The problem statement, all variables and given/known data
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.

2. Jan 10, 2010

### HallsofIvy

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.