@ by Sprmger-Verlag 1979 A Geometric Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators

tn this note we are concerned with unbounded self-adjoint operators in a Hilbert space. Denoting such an Operator by A we give a direct and geometric demonstration of the facts associated with the formula A = ~ 2dE(2).-oo 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 f 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'