Some functional analysis

**davidmccormick** · April 22nd 2010, 03:07 PM

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

Prove:

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

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

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

$(Af)(x) = \int_0^\pi sin(x-y)f(y)dy$
find the point spectrum and spectrum of A.

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

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

**Opalg** · April 23rd 2010, 12:02 AM

Originally Posted by davidmccormick

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

Prove:

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

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

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

$(Af)(x) = \int_0^\pi sin(x-y)f(y)dy$
find the point spectrum and spectrum of A.

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

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

For 1., you have dealt with the completeness requirement for a Hilbert space, but you have not explained the algebraic structure of the complexification. You need to define $\langle x+iy,z+iw\rangle = \langle x,z\rangle + \langle y,w\rangle + i\bigl(\langle y,z\rangle - \langle x,w\rangle\bigr)$ , for $x+iy$ and $z+iw$ in $H_{\mathbb{C}}$ , and show that this satisfies the axioms for a complex inner product.

Notice that that definition of the inner product implies that the norm on $H_{\mathbb{C}}$ is given by $\|x+iy\|^2 = \|x\|^2+\|y\|^2$ . That should help you with the problem that you are having with 2.

I don't offhand know the answer to 3. I'll try to think about that when I have a bit more time.

**Opalg** · April 23rd 2010, 11:38 PM

Originally Posted by davidmccormick

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

$(Af)(x) = \int_0^\pi sin(x-y)f(y)dy$
find the point spectrum and spectrum of A.

Let $c,\,s$ denote the elements of $L^2(o,\pi)$ given by $c(x) = \cos x$ , $s(x) = \sin x$ . Use the relation $\sin(x-y) = \sin x\cos y-\cos x\sin y$ to see that $Af = \langle f,c\rangle s - \langle f,s\rangle c$ . Thus A is a finite-rank operator, with two-dimensional range R spanned by $c$ and $s$ , and A is zero on the orthogonal complement of R. So the spectrum of A is equal to the point spectrum, which consists of 0 together with the spectrum of the operator A restricted to R.

Also, A restricted to R has matrix $\frac\pi2\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ (with respect to the basis consisting of $c$ and $s$ ), from which you can easily find the other two points in the spectrum.

**davidmccormick** · April 24th 2010, 12:07 PM

man you're a legend. I still can't show that B is bounded, but ur still a legend.

**davidmccormick** · April 24th 2010, 02:46 PM

The boundedness part also solved. thanks again.

Thread: Some functional analysis

LinkBack

Thread Tools

Display

Some functional analysis

solved

Similar Math Help Forum Discussions

Problem in functional analysis

functional analysis

Functional Analysis

problem in functional analysis

functional Analysis

Search Tags