On the existence of closed Hermitian restrictions of selfadjoint operators

Von Neumann [3, Satz 48] proved, by examining "the pathology of unbounded operators", that a closed hermitian operator H on a Hilbert space SC has a closed hermitian restriction if and only if it is unbounded. Hamburger [2, ?9] observed that von Neumann's argument did not show how to construct all such restrictions and that nothing was asserted about the deficiency index of the restriction. He then (loc. cit, Theorems 7 and 8) showed how to obtain all such restrictions, of deficiency index (m : m), with m finite or infinite, when H is selfadjoint (and unbounded, as appears in application of his method). We give here a brief proof of -von Neumann's result for the selfadjoint case: