2.3) R^DKj,. Definition 6.1. It will be convenient to use von Neumann's notation [2] A for the "closure" of A, i.e. A is the closed linear extension of A whose graph is the closure of the graph of A. Note 6.1. We always require that linear operators be single valued. Von Neumann [2] does admit more general operators so that some of the results stated by him would not be true in our convention, this applies most particularly to adjoints. It was seen in Part I that it is advantageous to use a

more »

... etrising operator whose null-space is as small as possible. The best we can achieve is given by LEMMA 6.1. Let H be a non-negative essentially self-adjoint operator which symmetrises A. Then a symmetrising operator can always be found which is self-adjoint and has as null-space the intersection of the closure of the range of A with the closure of the null-space of A. PROOF. Let H t = H+P where P is the orthogonal projector onto SRj;, the orthogonal complement of 3?^. Clearly H 1 is self-adjoint and H X A -HA since HA = HA because $ H D m A . Also for all / e 2)# (= 5) Hj ) so that H t is non-negative and satisfies the conditions of the lemma. It will be assumed in the future that the symmetrising operators H satisfy lemma 6.1. As foreshadowed in section 4 of our first paper [4] we shall have occasion to embed $, the domain and range space of our operators in a larger space § + § ' where §' is an exact replica of § but the elements of §' are orthogonal to the elements of §.