TITLE: Who first used the multiplication operator version of spectral theory
QUESTION [17 upvotes]: This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the functional calculus and that became more common after the Banach algebra revolution of the 40's.
Circa 1960 Dunford-Schwartz were still mainly resolutions of the identity.
By the time reed and I did Reed-Simon in the early 1970, we preferred to talk about multiplication operators emphasizing the individual spectral measures which was in the air when we wrote.  Who first using that language and who made it popular?

REPLY [3 votes]: Dieudonne records the following in Chapter VII of History of Functional Analysis:

Following the method of von Neumann, it is then easy to extend the homomorphism $f \mapsto f(N)$ to the algebra $\mathcal{U}(\sigma(N))$ of all universally measurable bounded functions in $\sigma(N)$.
  
  Finally, by adapting the arguments of von Neumann, Hellinger and Hahn, one arrives at the modern description of the Riesz-von Neumann "decomposition of unit" and of the "multiplicity theory" of Hellinger-Hahn:

Dieudonne then describes the multiplication version of Spectral Theory in 4 enumerated points, after which he writes:

One says that this description is a diagonalization of the normal operator $N$.

So Dieudonne describes diagonalization as a natural consequence of, application of, or interpretation of, the general theory up to that point. That would explain why there is no milestone to be found.