Celebratio Mathematica — Stone

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Works connected to Alberto González Domínguez

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M. H. Stone: “Some comments on the spectral theorem,” pp. 67–75 in Dedicado al Prof. Alberto González Domínguez [Dedicated to Prof. Alberto González Domínguez], published as Rev. Un. Mat. Argentina 25 : 1–2. Union Matematica Argentina (Bahia Blanca), 1970–1971. MR 0318945 Zbl 0238.47015 incollection

The spectral theorem for self-adjoint operators in a Hilbert space (with real, complex, or quaternionic scalars) generalizes the classical theorems on the canonical reduction of quadratic or hermitian forms and their matrices. Usually two steps are needed, the first passing from finite-dimensional spaces to bounded operators in general spaces, the second from bounded operators to unbounded. There has always been a certain interest (see [Riesz 1930; Lengyel and Stone 1936; Tseng 1935; Bernau 1968; Wouk 1966]) in carrying out this generalization by “pure” Hilbert space methods–that is to say, by using only intrinsic algebraic and geometric properties of abstract Hilbert space without recourse to special theorems drawn from classical analysis. For bounded operators the spectral theorem was treated in this spirit by F. Riesz [1930] and by Lengyel and Stone [1936], for unbounded operators by Y. Y. Tseng [1935]. The present paper, while closely related to Tseng’s, expounds a variant of his approach that may appear somewhat simpler and may shed some additional light on the techniques required.