Book: Functional Calculus

Question

Is there a good book that investigates in detail the various kinds of functional calculus?

I'm having now some knowledge about unbounded operators and integration but I would like to understand better functional calculus especially in order to prove Stone's Theorem.

What I'm not looking for is an approach to functional calculus via the spectral theorem.

Answer 1

The two books that I recommend are:

• John B. Conway’s A Course in Functional Analysis.
• Jiří Blank’s, Pavel Exner’s and Miloslav Havlíček’s Hilbert Space Operators in Quantum Physics.

Conway’s book includes a detailed proof of Stone’s Theorem using spectral theory. (For your information, spectral theory is not the only approach to Stone’s Theorem. It can also be proven using the group $ C^{*} $-algebra $ {C^{*}}(\mathbb{R}) $ and some Fourier analysis.) The main highlight of the book, however, is using spectral theory to prove the following theorem:

Theorem: If $ N $ is a bounded normal operator on a Hilbert space $ \mathcal{H} $, then there is a measure space $ (X,\Sigma,\mu) $ and a function $ \phi \in {L^{\infty}}(X,\Sigma,\mu) $ such that $ N $ is unitarily equivalent to the multiplication operator $ M_{\phi} $ on $ {L^{2}}(X,\Sigma,\mu) $.

Conway claims that this theorem is the optimal form of the spectral theorem for normal bounded operators on a Hilbert space.

The book by the three Czech physicists contains surprisingly rigorous mathematics, and I recommend it because it connects spectral theory with John von Neumann’s attempt to address foundational issues in quantum mechanics. There is a rigorous discussion of Stone’s Theorem and of the time-evolution problem in quantum mechanics (even for a time-dependent Hamiltonian, which is tackled using the Dyson expansion formula).

Answer 2

As a suggestion of reading...

Operator Theory: Weidmann

Spectral Theory: Birman & Solomjak

Semigroup Theory: Engel & Nagel