Let

be a real Hilbert space. Define the complexification of

by

=

.

Prove:

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.