How Does Trace Class Operator Q Relate to Norms in Hilbert Spaces?

[camillio]

Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof.

He states in Lemma 1.1.4:
Let μ be a finite Borel measure on H. Then the following assertions are equivalent:
(1) $\int_H |x|^2 \mu(dx) < \infty$
(2) There exists a positive, symmetric, trace class operator Q s.t. for $x,y \in H$
$$<Qx, y> = \int_H <x,z><y,z> \mu(dz).$$

If (2) holds, then $Tr Q = \int_H |x|^2 \mu(dx)$.
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The proof begins:
Spse (2) holds. Let $(e_n)_{n\in \mathbb{N}}$ be an orthonormal basis of H. Then
$$\int_H |x|^2 \mu(dx) = \sum_{n=1}^\infty \int_H |<x, e_n>|^2 \mu(dx) = \sum_{n=1}^\infty <Qe_n, e_n> = Tr Q < \infty.$$

What I have trouble with is the transitiono to the sum of $<Qe_n, e_n>$. If I suppose, that $x, e_n$ may be complex, then I miss the adjoint part of the absolute value.

Most probably I miss some trivial notion, so any help will be appreciated.

 

[voko]

By definition, $|z|^2 = z\overline{z}$, so $|<x, e_n>|^2 = <x, e_n>\overline{<x, e_n>} = <x, e_n><e_n, x>$.

 

[camillio]

I know, but what then with the following?
$$<Qx, y> = \int_H <x,z> <y,z> \mu(dz)$$

If I understand correctly,
$$<Qe_n, e_n> = \int <e_n, x> <e_n, x> \mu(dx) = \int <e_n, x>^2 \mu(dx)$$
which doesn't coincide with abs. value for complex numbers.

 

[voko]

Read the first paragraph in 1.1. H is a real Hilbert space.

 

[camillio]

Damn, you're right! I'm deeply sorry, my trivial fault :-(

 

[dextercioby]

I'm sorry, can you, please, post a link to the book ? I couldn't find it on google books either by name, or by title...

Thanks! (later edit).

 

[voko]

http://siba-ese.unisalento.it/index.php/quadmat/issue/view/775