# Spectral Measures: References

I am trying to learn a little bit about the spectral theory of unbounded operators but the textbook we are using (Birman and Solomyak: Spectral theory of Self-Adjoint Operators in a Hilbert Space) is a little bit heavy going for me. Is there a "gentler" book to learn about these things?

Thank you.

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Have you learned about the spectral theory of bounded operators? For those not familiar with Birman and Solomyak or your background, could you please elaborate a little on what would help make a book "gentle" enough in this context? – Jonas Meyer Nov 11 '11 at 2:51
I'll just briefly add a comment (because I cannot tell if it will be helpful) that I like Conway's treatment in section X.4 of A course on functional analysis. It builds on some previous results, including the spectral theory of bounded normal operators from Chapter IX. – Jonas Meyer Nov 11 '11 at 2:58
Thanks for your comments. I don't fully understand the bounded case( i.e I am just beginning to study functional analysis and perhaps should get back to this topic a bit later). I will check out Conway's book. Thanks for the reference pointer. I suppose my question can be paraphrased as follows: is there another book that discusses spectral measures and the spectral theorem based on the spectral measure's and projection valued functions. Thanks. – Shibi Vasudevan Nov 11 '11 at 22:07

I like Conway's treatment in section X.4 of A course on functional analysis. It builds on some previous results, including the spectral theory of bounded normal operators from Chapter IX.

I don't know if it is gentler.

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A very good reference is Schmüdgen: "Unbounded Self-adjoint Operators on Hilbert Space", if you want to focus on Hilbert spaces.

http://www.springer.com/en/book/9789400747524

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References

Operator theoretic treatments:

• William Arveson (Spetral Measures, Pettis Integral)*
• M. S. Birman, M. Z. Solomjak (Spectral Measures, Pettis Integral)
• J. Blank, P. Exner, M. Havlíček (Spectral Measures, Pettis Integral)
• John B. Conway (Spectral Measures, Pettis Integral)
• Alexander Frei (Spectral Measures, Pettis Integral)
• Emmanuel Kowalski (Spectral Measures, Pettis Integral)
• Konrad Schmüdgen (Spectral Families, Stiltjies Integral)

(This list is not comprehensive!)

Recommendations

Recommendations with highlights:

• M. S. Birman, M. Z. Solomjak (Book): Extension of Pettis Integral
• John B. Conway (Book): Approximation of Normals Operators
• Alexander Frei (Posts): Transformation of Normal Operators

(This list is personal opinion.)

Denominations

By spectral measures it is meant maps from Borel algebra.

By spectral families it is meant maps from real line.

Attention

Spectral families and Stiltjies integrals are superflous:

They seemed to arose for missing interpretion of integrals.

The modern approach exploits the Pettis integral.

(In fact, it only imitates the Pettis integral.)

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