Finding the Adjoint of an Infinite Hilbert Space Operator

[Niles]

Homework Statement

Hi guys

I have the following operator on an infinite Hilbert space: Oe_i=e_i+1, where i is 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.

 

[HallsofIvy]

You've done the hard work, that's trivial now! For any basis vector e_i, O*e_i= e_{i_1} so OO*e_i= Oe_i-1= e_(i-1)+1= e_i.

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