Let be a real Hilbert space. Define the complexification of by = .
1. is a Hilbert space
2. Define where is a bounded operator on . Prove that B is bounded on H_C.
3. Let be defined by:
find the point spectrum and spectrum of A.
For 1. I say, suppose is a Cauchy sequence in . Then are Cauchy in H and since H is complete converge to some x,y in H. Therefore, which is in so it's complete. Is this ok?
For 2. I can only show that where which is not good enough as it should be M (||x+iy||) to fit the definition of a bounded operator. So please help.
For 3. sadly I have no clue.
thanks for your help.