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.