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**davidmccormick** Let $\displaystyle H$ be a real Hilbert space. Define the complexification of $\displaystyle H$ by $\displaystyle H_{\mathbb{C}}$=$\displaystyle \{x+iy: x,y \in H\}$.

Prove:

1. $\displaystyle H_{\mathbb{C}}$ is a Hilbert space

2. Define $\displaystyle B(x+iy) = A (x) + i A(y)$ where $\displaystyle A$ is a bounded operator on $\displaystyle H$. Prove that B is bounded on H_C.

3. Let $\displaystyle A:{L^2}(0,\pi)\longrightarrow{L^2}(0,\pi)$ be defined by:

$\displaystyle (Af)(x) = \int_0^\pi sin(x-y)f(y)dy$

find the point spectrum and spectrum of A.

For 1. I say, suppose $\displaystyle (z_n)=(x_n + iy_n)$ is a Cauchy sequence in $\displaystyle H_{\mathbb{C}}$. Then $\displaystyle (x_n), (y_n)$ are Cauchy in H and since H is complete converge to some x,y in H. Therefore, $\displaystyle z_n\longrightarrow{z}=x+iy$ which is in $\displaystyle H_{\mathbb{C}}$ so it's complete. Is this ok?

For 2. I can only show that $\displaystyle ||B(w)||_{op} \leq M (||x|| + ||y|| )$ where $\displaystyle w = x + iy$ 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.