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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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.


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)
  • Brian C. Hall (Spectral Measures, Pettis Integral, Direct 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 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.)


By spectral measures it is meant maps from Borel algebra.

By spectral families it is meant maps from real line.


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.)

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