I would like to learn about the spectral theory of unbounded operators. I'm looking for a lucid, rigorous, self-contained and basic exposition of this topic that assumes no more than the material covered in Rynne and Youngson's "Linear Functional Analysis". It can be in the form of a book, class notes, an article, etc. Language can be English, French or German. Thank you.
I prefer sticking to the classical context for the first round of dealing with the spectral theorem; in particular, I would use Riemann-Stieltjes integrals instead of Borel measures. Once you have the Riemann-Stieltjes version, it is a fairly trivial matter to extend to the measure theoretic, when you want it. I highly recommend this text for self-study at your level.
George Bachman and Lawrence Narici
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I was able to teach myself spectral theory as an undergraduate from this text, and I knew others who did the same. The authors supply three different proofs of the Spectral Theorem for the bounded case, each with a different slant. And they offer two different proofs for the unbounded case! You can take your pick because the proofs are independent of each other.
The book is long because the Authors go to great lengths to make the exposition clear and available to someone is learning on their own--and it is very clear. This text will build naturally on what you already know, and I think you'll come away feeling that you've learned something substantial, and not just a bunch of tricks. The exercises are great.
I asked about Complex Analysis because I wanted to find a proof using Complex Analysis, but all the presentations that I can find introduce far too much that isn't really necessary. That will just bog you down instead of helping.