Who first proposed the idea of "resolution of the identity" as used in the functional calculus of self-adjoint operators? Was it von Neumann?

In Japanese, it translates as "resolution of the unit". To figure out why it was translated that way, I did some research and now I'd like to know who came up with the idea in the first place.

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    $\begingroup$ Could you provide additional context? What does that mean? $\endgroup$
    – Mauricio
    Oct 10, 2021 at 9:16

1 Answer 1


It was Hilbert's way to generalize spectral theory to infinite dimensional operators when he discovered continuous spectrum in 1906. Hilbert used bilinear forms when writing integrals, and the more recognizable operator version was introduced by Riesz in Les systemes d'equations lineaires a une infinite d'inconnues (1913), see Dieudonne, History of functional analysis, VII.2. The name "resolution of the identity" was popularized by Stone in 1930s, see Steen, Highlights in the History of Spectral Theory.

Hilbert and Riesz only dealt with bounded operators, Stone and von Neumann extended the theory to unbounded self-adjoint operators in 1930s. Riesz's form of the spectral theorem remained dominant until around 1950 (according to Dieudonne). After the study of Banach algebras in 1940s, it started to be gradually eclipsed by the multiplicative form spelled out in passing by Stone in his 1932 book (based on von Neumann's ideas from a year before), see MathOverflow, Who first used the multiplication operator version of spectral theory.


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