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

In summary, the conversation discusses a proof in the book "Gaussian measures on Hilbert spaces" by S. Maniglia. The proof involves showing the equivalence between two assertions, and using an orthonormal basis and a trace class operator. There is some confusion regarding the use of complex numbers in the proof, but it is clarified that the Hilbert space in question is real. A link to the book is also provided.
  • #1
camillio
74
2
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) [itex] \int_H |x|^2 \mu(dx) < \infty[/itex]
(2) There exists a positive, symmetric, trace class operator Q s.t. for [itex]x,y \in H[/itex]
[tex] <Qx, y> = \int_H <x,z><y,z> \mu(dz).[/tex]

If (2) holds, then [itex]Tr Q = \int_H |x|^2 \mu(dx)[/itex].
--------

The proof begins:
Spse (2) holds. Let [itex](e_n)_{n\in \mathbb{N}}[/itex] be an orthonormal basis of H. Then
[tex]
\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.
[/tex]

What I have trouble with is the transitiono to the sum of [itex]<Qe_n, e_n>[/itex]. If I suppose, that [itex]x, e_n[/itex] 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.
 
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  • #2
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> ##.
 
  • #3
I know, but what then with the following?
[tex]
<Qx, y> = \int_H <x,z> <y,z> \mu(dz)
[/tex]

If I understand correctly,
[tex]
<Qe_n, e_n> = \int <e_n, x> <e_n, x> \mu(dx) = \int <e_n, x>^2 \mu(dx)
[/tex]
which doesn't coincide with abs. value for complex numbers.
 
  • #4
Read the first paragraph in 1.1. H is a real Hilbert space.
 
  • #5
Damn, you're right! I'm deeply sorry, my trivial fault :-(
 
  • #6
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).
 
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FAQ: How Does Trace Class Operator Q Relate to Norms in Hilbert Spaces?

What is the norm in Hilbert space?

The norm in Hilbert space is a mathematical concept used to measure the size or length of a vector in a Hilbert space. It is a generalization of the concept of length or magnitude of a vector in Euclidean space and is defined as the square root of the inner product of a vector with itself.

How is the norm calculated in Hilbert space?

The norm in Hilbert space is calculated by taking the square root of the inner product of a vector with itself. This inner product is a measure of the angle between two vectors and is defined as the sum of the products of the corresponding components of the vectors.

What is the significance of the norm in Hilbert space?

The norm in Hilbert space has several important applications in mathematics and physics. It is used to define the distance between vectors, to measure the convergence of sequences of vectors, and to define the concept of orthogonality. It is also used in the formulation of optimization problems and in the study of linear operators.

How is the norm related to the concept of convergence in Hilbert space?

The norm in Hilbert space is closely related to the concept of convergence. In particular, a sequence of vectors in a Hilbert space is said to converge to another vector if the norm of the difference between the two vectors approaches zero as the number of terms in the sequence increases. This allows us to define the limit of a sequence of vectors and to study their properties.

Can the norm in Hilbert space be used in other types of spaces?

Yes, the concept of norm can be extended to other types of spaces such as Banach spaces and topological vector spaces. In these spaces, the norm is used to define the distance between vectors and to study the convergence of sequences. However, the properties of the norm may differ in these different types of spaces.

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