A geometric proof of the spectral theorem for unbounded self-adjoint operators

Abstract

That is, we establish directly the well-known spectral theorem for unbounded self-adjoint operators using only simple geometric intrinsic properties of Hilbert space. Of course the fundamental facts about the spectral representation of bounded as well as unbounded operators have been known in substance since the appearance in 1906 [3] of Hilbert's memoir on integral equations and in 1929 [8] of von Neumann's fundamental paper on unbounded operators. Since that time many papers have been published in this subject using a variety of methods. Some of these methods apply only to bounded operators, while others are suited to the general (unbounded) case. However, all but a few of these methods use techniques and principles which lie outside of Hilbert space theory proper, such as Helly's selection principle, Riesz's second representation theorem and so on. For further examples of technique and for a comprehensive list of references on the spectral theorem we refer the reader to [1], p. 927 and [5]. It was not until 1935 that Lengyel and Stone [5] gave a new proof of the spectral theorem which was strictly elementary in the sense that it depended only upon intrinsic properties of Hilbert space. Their paper dealt with the case of bounded operators and the authors remarked that they could not handle the general case in the same way. In their introduction they wrote: "Indeed our method, which requires the study of powers of an operator, is not suited to the case of unbounded operators" ([5], p. 853). Just this sentence stimulated the author to try to handle the general case in the same way and indeed it is possible. The fundamental idea of our proof of the spectral theorem as well as that of Lengyel and Stone consists in considering invariant subspaces F(A,2)= {xenlxeD(A"), ]lA"xl[ <2" . Itxll for n = 1,2, 3 .... }.