If then is invertible and so the only possible eigenvector of would be . For the case I don't know how to prove this yet.
is a Hilbert space and is such that . Show that .
I have tried looking at this problem in several ways and I just cannot seem to see the significance of nor how to go about proving it. I understand how the adjoint is defined for a Hilbert space as a consequence of Reisz representation. I have tried to attack it using fixed point theorems. Any nudge in the right direction would be appreciated.