Remind me where to find a proof of a spectral theorem for RHS

In summary: In any case, it looks like he only wrote the introduction and conclusion of the document. The article I'm thinking of is about a spectral theorem that is valid for rigged Hilbert spaces. The proof is pretty short, and I don't think it goes into too much detail. I think the main thing to take away from it is that this spectral theorem is valid for spaces that are not necessarily commutative.
  • #1
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,877
422
I know I've seen a very short article (3-5 pages) with a proof of a spectral theorem for rigged Hilbert spaces, written for people who already know the usual spectral theorems, by some guy who I think had a muslim name. Anyone know what I'm talking about? It's been posted here before, almost certainly more than once, but I just spent 20 minutes trying to find it here and on my computer, so I think it's time to just ask.

No need to hurry to get me an answer right away. I'm not even going to read it right away. I'm just making a to-do list about things I'd like to understand better, and I want to put this article on the list.

(I think I may even have asked this question before, LOL).
 
Physics news on Phys.org
  • #2
You can always try Gelfand-Vilenkin's Generalized functions, Volume 4. However, the proof there has a mistake, which is not easily solved (but can be solved).
My favorite book is "Methods of Hilbert spaces" by Maurin. It states the proof in a different form than Gelfand-Vilenkin, and a form that I think is more useful. Than again, he doesn't explicitely say anything about RHS (he obviously does use the concept though). Other than that, the book really covers a lot of nice things about Hilbert spaces. I don't understand why the book is not more popular.
 
  • #3
Thanks micro. I'll make a note of those books before I forget them too. It's interesting to hear that you liked Maurin's book. One of the reasons why I haven't checked it out is that I found a negative review about it, that complained about the presentation, inconsistencies in notation, and even some incorrect statements. Link.

I still suspect that the short article I've seen is the best place to study the theorem and its proof. I just wish I could remember. I hope that Strangrep or dextercioby does.
 
  • #4
I found the document I had in mind. It's a 5-page pdf document with the title "Generalized eigenfunctions" written by a guy named Mustafa Kesir. Some other guy named Christopher King has a copy on his web page: http://mathserver.neu.edu/~king_chris/GenEf.pdf.
 
  • #5
I remember saying sometimes in the past that Kesir almost copy-pasted in his work one of the appendices of Berezin & Shubin's 'Schrödinger equation' which had been based on Berezanskii's work and book in the 1960's.
 

1. What is the spectral theorem for RHS?

The spectral theorem for RHS (right-hand side) is a mathematical theorem that states that every compact self-adjoint operator on a Hilbert space has a spectral decomposition. This means that it can be expressed as a linear combination of projections onto the eigenspaces of the operator.

2. Why is the spectral theorem important?

The spectral theorem is important because it provides a way to decompose a complicated operator into simpler components, making it easier to study and understand. It also has many applications in areas such as quantum mechanics, signal processing, and differential equations.

3. What is the proof of the spectral theorem for RHS?

The proof of the spectral theorem for RHS involves using the properties of compact self-adjoint operators and the spectral theorem for normal operators. It also relies on the properties of the eigenvectors and eigenvalues of the operator. A detailed proof can be found in most advanced linear algebra or functional analysis textbooks.

4. Where can I find a proof of the spectral theorem for RHS?

A proof of the spectral theorem for RHS can be found in many advanced mathematics textbooks, particularly those focused on linear algebra or functional analysis. It can also be found in various online resources, such as research papers and lecture notes.

5. Can the spectral theorem for RHS be applied to non-compact operators?

No, the spectral theorem for RHS only applies to compact operators. However, there are generalizations of the theorem for non-compact operators, such as the continuous and integral spectral theorems, which can be found in more advanced texts on functional analysis.

Similar threads

I Spectral theorem for Hermitian matrices-- special cases
    • Linear and Abstract Algebra
Replies
2
Views
618
B Question about the fundamental theorem of calculus
    • Calculus
Replies
2
Views
307
I Virial theorem as applied to hydrogen atom
    • Quantum Physics
Replies
2
Views
5K
B Bell's theorem, QFT, and the Relativity of Simultaneity
    • Quantum Physics
2
Replies
64
Views
4K
I Epistemic view of the wave function leads to superluminal signal
    • Quantum Physics
2
Replies
41
Views
3K
I Noether's second theorem: two questions
    • Special and General Relativity
Replies
7
Views
1K
B A proof of the fundamental theorem of calculus
    • Calculus
Replies
6
Views
2K
Proof of Gelfand-Maurin Nuclear Spectral Theorem?
    • Quantum Physics
Replies
5
Views
7K
Busch's Gleason-like theorem
    • Quantum Physics
2
Replies
54
Views
10K
Wiener-Khinchin theorem and time delay
    • Quantum Physics
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top